Haddad, Wassim M.; Bernstein, Dennis S.
1991-01-01
Lyapunov function proofs of sufficient conditions for asymptotic stability are given for feedback interconnections of bounded real and positive real transfer functions. Two cases are considered: (1) a proper bounded real (resp., positive real) transfer function with a bounded real (resp., positive real) time-varying memoryless nonlinearity; and (2) two strictly proper bounded real (resp., positive real) transfer functions. A similar treatment is given for the circle and Popov theorems. Application of these results to robust stability with time-varying bounded real, positive real, and sector-bounded uncertainty is discussed.
Haddad, Wassim M.; Bernstein, Dennis S.
1991-01-01
Lyapunov function proofs of sufficient conditions for asymptotic stability are given for feedback interconnections of bounded real and positive real transfer functions. Two cases are considered: (1) a proper bounded real (resp., positive real) transfer function with a bounded real (resp., positive real) time-varying memoryless nonlinearity; and (2) two strictly proper bounded real (resp., positive real) transfer functions. A similar treatment is given for the circle and Popov theorems. Application of these results to robust stability with time-varying bounded real, positive real, and sector-bounded uncertainty is discussed.
Analysis of stability problems via matrix Lyapunov functions
Directory of Open Access Journals (Sweden)
Anatoly A. Martynyuk
1990-01-01
Full Text Available The stability of nonlinear systems is analyzed by the direct Lyapunov's method in terms of Lyapunov matrix functions. The given paper surveys the main theorems on stability, asymptotic stability and nonstability. They are applied to systems of nonlinear equations, singularly-perturbed systems and hybrid systems. The results are demonstrated by an example of a two-component system.
Inertia theorems for operator Lyapunov inequalities
Sasane, AJ; Curtain, RF
2001-01-01
We study operator Lyapunov inequalities and equations for which the infinitesimal generator is not necessarily stable, but it satisfies the spectrum decomposition assumption and it has at most finitely many unstable eigenvalues. Moreover, the input or output operators are not necessarily bounded, bu
Inertia theorems for operator Lyapunov inequalities
Sasane, AJ; Curtain, RF
2001-01-01
We study operator Lyapunov inequalities and equations for which the infinitesimal generator is not necessarily stable, but it satisfies the spectrum decomposition assumption and it has at most finitely many unstable eigenvalues. Moreover, the input or output operators are not necessarily bounded,
A Lyapunov approach to strong stability of semigroups
Paunonen, L.T.; Zwart, Heiko J.
2013-01-01
In this paper we present Lyapunov based proofs for the well-known Arendt–Batty–Lyubich–Vu Theorem for strongly continuous and discrete semigroups. We also study the spectral properties of the limit isometric groups used in the proofs.
On stability of discontinuous systems via vector Lyapunov functions
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
This paper deals with the stability of systems with discontinuous righthand side (with solutions in Filippov's sense) via locally Lipschitz continuous and regular vector Lyapunov functions. A new type of "set-valued derivative" of vector Lyapunov functions is introduced, some generalized comparison principles on dis(c)ontinuous systems are shown. Furthermore, Lyapunov stability theory is developed for a class of discontinuous systems based on locally Lipschitz continuous and regular vector Lyapunov functions.
Stabilization of the Ball on the Beam System by Means of the Inverse Lyapunov Approach
Directory of Open Access Journals (Sweden)
Carlos Aguilar-Ibañez
2012-01-01
Full Text Available A novel inverse Lyapunov approach in conjunction with the energy shaping technique is applied to derive a stabilizing controller for the ball on the beam system. The proposed strategy consists of shaping a candidate Lyapunov function as if it were an inverse stability problem. To this purpose, we fix a suitable dissipation function of the unknown energy function, with the property that the selected dissipation divides the corresponding time derivative of the candidate Lyapunov function. Afterwards, the stabilizing controller is directly obtained from the already shaped Lyapunov function. The stability analysis of the closed-loop system is carried out by using the invariance theorem of LaSalle. Simulation results to test the effectiveness of the obtained controller are presented.
Lyapunov functionals and stability of stochastic functional differential equations
Shaikhet, Leonid
2013-01-01
Stability conditions for functional differential equations can be obtained using Lyapunov functionals. Lyapunov Functionals and Stability of Stochastic Functional Differential Equations describes the general method of construction of Lyapunov functionals to investigate the stability of differential equations with delays. This work continues and complements the author’s previous book Lyapunov Functionals and Stability of Stochastic Difference Equations, where this method is described for discrete- and continuous-time difference equations. The text begins with a description of the peculiarities of deterministic and stochastic functional differential equations. There follow basic definitions for stability theory of stochastic hereditary systems, and a formal procedure of Lyapunov functionals construction is presented. Stability investigation is conducted for stochastic linear and nonlinear differential equations with constant and distributed delays. The proposed method is used for stability investigation of di...
Stabilization of nonlinear systems based on robust control Lyapunov function
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CAI Xiu-shan; HAN Zheng-zhi; LU Gan-yun
2007-01-01
This paper deals with the robust stabilization problem for a class of nonlinear systems with structural uncertainty. Based on robust control Lyapunov function, a sufficient and necessary condition for a function to be a robust control Lyapunov function is given. From this condition, simply sufficient condition for the robust stabilization (robust practical stabilization) is deduced. Moreover, if the equilibrium of the closed-loop system is unique, the existence of such a robust control Lyapunov function will also imply robustly globally asymptotical stabilization. Then a continuous state feedback law can be constructed explicitly. The simulation shows the effectiveness of the method.
Stability of time-delay systems via Lyapunov functions
Directory of Open Access Journals (Sweden)
Carlos F. Alastruey
2002-01-01
Full Text Available In this paper, a Lyapunov function candidate is introduced for multivariable systems with inner delays, without assuming a priori stability for the nondelayed subsystem. By using this Lyapunov function, a controller is deduced. Such a controller utilizes an input–output description of the original system, a circumstance that facilitates practical applications of the proposed approach.
Lyapunov functionals and stability of stochastic difference equations
Shaikhet, Leonid
2011-01-01
This book offers a general method of Lyapunov functional construction which lets researchers analyze the degree to which the stability properties of differential equations are preserved in their difference analogues. Includes examples from physical systems.
Stabilization of discrete nonlinear systems based on control Lyapunov functions
Institute of Scientific and Technical Information of China (English)
无
2008-01-01
The stabilization of discrete nonlinear systems is studied.Based on control Lyapunov functions,asufficient and necessary condition for a quadratic function to be a control Lyapunov function is given.From this condition,a continuous state feedback law is constructed explicitly.It can globally asymptotically stabilize the equilibrium of the closed-loop system.A simulation example shows the effectiveness of the proposed method.
Can stability analysis be really simplified? (revisiting Lyapunov, Barbalat, LaSalle and all that)
Barkana, Itzhak
2017-01-01
Even though Lyapunov approach is the most commonly used method for stability analysis, its use has been hindered by the realization that in most applications the so-called Lyapunov derivative is at most negative semidefinite and not negative definite as desired. Many different approaches have been used in an attempt to overcome these difficulties. Until recently, the most widely accepted stability analysis has been based on Barbalat's Lemma which seems to require uniform continuity of practically all signals involved. Recently, stability analysis methods for nonautonomous nonlinear systems have been revisited. Even though new developments based on unknown works of LaSalle attempted to mitigate these continuity conditions, counterexamples are suggested to contradict these results. New analysis shows that these counterexamples, which are making use of well-known mathematical expressions, are actually using them beyond their domain of validity. Therefore, the restrictive condition of uniform continuity required by Barbalat's Lemma and even the milder conditions required by LaSalle's extension of the Invariance Principle to nonautonomous systems can be further mitigated. A new Invariance Principle only required that bounded trajectories cannot pass an infinite distance in finite time. Finally, a new Theorem of Stability, which is formulated as a direct extension and a generalization of Lyapunov's Theorem, not only simplifies the stability analysis of nonlinear systems, but also leads to conclusive results about the system under analysis.
Circular orbits, Lyapunov stability and Manev-type forces
Blaga, Cristina
2016-01-01
In this article we study the stability in the sense of Lyapunov of the circular orbits in the generalized Manev two bodies problem. First, we explore the existence of the circular orbits and determine their radius. Then, using the first integrals of motion we build a positive definite function, known as a Lyapunov function. It's existence proves that the circular orbit is stable in the sense of Lyapunov. In the end, we consider several real systems of two bodies and compare the characteristics of the circular orbits in Newtonian and modified Manev gravitational field, arguing about our possibilities to observe the differences between the motion in these two fields.
On Brayton and Moser's missing stability theorem
Jeltsema, D.; Scherpen, J. M. A.
2005-01-01
In the early 1960s, Brayton and Moser proved three theorems concerning the stability of nonlinear electrical circuits. The applicability of each theorem depends on three different conditions on the type of admissible nonlinearities in circuit. Roughly speaking, this means that the theorems apply to
Minimal Generalized Lyapunov Function Theorems for Nonlinear Systems%非线性系统的最小广义Lyapunov函数定理
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玉强; 吴保卫
2014-01-01
利用Zorn引理和LaSalle不变原理，研究一般非线性系统最小广义 Lyapunov 函数的存在性，证明了最小(非负)广义 Lyapunov 函数和最小(非负)强广义 Lyapunov 函数的存在性定理。%Based on Zorn’s lemma and LaSalle’s invariance principle,the existence of the minimal generalized Lyapunov function for nonlinear systems was considered. We developed the minimal generalized Lyapunov function theorems and the minimal strong generalized Lyapunov function theorems for nonlinear dynamical systems.
Control Lyapunov Stabilization of Nonlinear Systems with Structural Uncertainty
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CAI Xiu-shan; HAN Zheng-zhi; TANG Hou-jun
2005-01-01
This paper deals with global stabilization problem for the nonlinear systems with structural uncertainty.Based on control Lyapunov function, a sufficient and necessary condition for the globally and asymptotically stabilizing the equailibrium of the closed system is given. Moreovery, an almost smooth state feedback control law is constructed. The simulation shows the effectiveness of the method.
STABILIZATION OF NONLINEAR TIME-VARYING SYSTEMS: A CONTROL LYAPUNOV FUNCTION APPROACH
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Zhongping JIANG; Yuandan LIN; Yuan WANG
2009-01-01
This paper presents a control Lyapunov function approach to the global stabilization problem for general nonlinear and time-varying systems. Explicit stabilizing feedback control laws are proposed based on the method of control Lyapunov functions and Sontag's universal formula.
Lyapunov Criteria for Structural Stability of Supply Chain System
Institute of Scientific and Technical Information of China (English)
LU Ying-jin; TANG Xiao-wo; ZHOU Zong-fang
2004-01-01
In this paper, based on Cobb-Douglas production function, the structural stability of the supply chain system are analyzed by employing Lyapunov criteria. That the supply chain system structure,with the variance of the rate of re-production input funding, becomes unstable is proved. Noticeably, the solutions shows that when the optimal combination of input parameter element, the qualitative properties of supply chain system change and the supply chain system becomes unstable.
2015-01-01
The objective of this paper is to study detectability, observability and related Lyapunov-type theorems of linear discrete-time time-varying stochastic systems with multiplicative noise. Some new concepts such as uniform detectability, ${\\cal K}^{\\infty}$-exact detectability (resp. ${\\cal K}^{WFT}$-exact detectability, ${\\cal K}^{FT}$-exact detectability, ${\\cal K}^{N}$-exact detectability) and ${\\cal K}^{\\infty}$-exact observability (resp. ${\\cal K}^{WFT}$-exact observability, ${\\cal K}^{FT}...
Control design and comprehensive stability analysis of acrobots based on Lyapunov functions
Institute of Scientific and Technical Information of China (English)
LAI Xu-zhi; WU Yun-xin; SHE Jin-hua; WU Min
2005-01-01
A design method for controllers and a comprehensive stability analysis for an acrobat based on Lyapunov functions are presented. Three control laws based on three Lyapunov functions are designed to increase the energy so as to move the acrobot into the unstable inverted equilibrium position, and solve the problem of posture and energy. The concept of a non-smooth Lyapunov function is employed to analyze the stability of the whole system. The validity of this strategy is demonstrated by simulations.
2013-06-01
STABILITY BY COMPUTING A SINGLE QUADRATIC LYAPUNOV FUNCTION Mehrdad Pakmehr∗ PhD Candidate Decision and Control Laboratory (DCL) School of Aerospace...linearization and linear matrix inequality (LMI) techniques. Using convex optimization tools, a single quadratic Lyapunov function is computed for multiple...Scheduling Control of Gas Turbine Engines: Stability by Computing a Single Quadratic Lyapunov Function 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c
Global stabilization of nonlinear systems based on vector control lyapunov functions
Karafyllis, Iasson
2012-01-01
This paper studies the use of vector Lyapunov functions for the design of globally stabilizing feedback laws for nonlinear systems. Recent results on vector Lyapunov functions are utilized. The main result of the paper shows that the existence of a vector control Lyapunov function is a necessary and sufficient condition for the existence of a smooth globally stabilizing feedback. Applications to nonlinear systems are provided: simple and easily checkable sufficient conditions are proposed to guarantee the existence of a smooth globally stabilizing feedback law. The obtained results are applied to the problem of the stabilization of an equilibrium point of a reaction network taking place in a continuous stirred tank reactor.
Lyapunov functions and global stability for SIR and SEIR models with age-dependent susceptibility
Korobeinikov, Andrei
2013-01-01
We consider global asymptotic properties for the SIR and SEIR age structured models for infectious diseases where the susceptibility depends on the age. Using the direct Lyapunov method with Volterra type Lyapunov functions, we establish conditions for the global stability of a unique endemic steady state and the infection-free steady state.
Lyapunov functions and global stability for SIR and SEIR models with age-dependent susceptibility.
Melnik, Andrey V; Korobeinikov, Andrei
2013-04-01
We consider global asymptotic properties for the SIR and SEIR age structured models for infectious diseases where the susceptibility depends on the age. Using the direct Lyapunov method with Volterra type Lyapunov functions, we establish conditions for the global stability of a unique endemic steady state and the infection-free steady state.
Stability, Resonance and Lyapunov Inequalities for Periodic Conservative Systems
Canada, Antonio
2010-01-01
This paper is devoted to the study of Lyapunov type inequalities for periodic conservative systems. The main results are derived from a previous analysis which relates the best Lyapunov constants to some especial (constrained or unconstrained) minimization problems. We provide some new results on the existence and uniqueness of solutions of nonlinear resonant and periodic systems. Finally, we present some new conditions which guarantee the stable boundedness of linear periodic conservative systems.
Broucke, R.
1982-01-01
It is pointed out that the Lyapunov Characteristic Numbers constitute a new tool for determining stability of trajectories of dynamical systems, or, even more generally, of solutions of systems of ordinary differential equations. In contrast with the characteristic exponents, which apply only to periodic solutions, the Lyapunov Characteristic Numbers apply to arbitrary nonperiodic solutions as well. A description is presented of the numerical experiments which have been made in order to investigate the practical value of the Lyapunov Characteristic Number and the Kolmogorov Entropy for the purpose of estimating the stability of trajectories and/or numerical integration methods in celestial mechanics. It is found that the Lyapunov Characteristic Numbers are extremely useful for the classification of the solutions of nonintegrable dynamical systems, especially in order to distinguish between quasi-periodic and chaotic solutions. However, the Lyapunov Characteristics Numbers do not appear to be useful for the purpose of evaluating numerical integration methods.
Broucke, R.
1982-01-01
It is pointed out that the Lyapunov Characteristic Numbers constitute a new tool for determining stability of trajectories of dynamical systems, or, even more generally, of solutions of systems of ordinary differential equations. In contrast with the characteristic exponents, which apply only to periodic solutions, the Lyapunov Characteristic Numbers apply to arbitrary nonperiodic solutions as well. A description is presented of the numerical experiments which have been made in order to investigate the practical value of the Lyapunov Characteristic Number and the Kolmogorov Entropy for the purpose of estimating the stability of trajectories and/or numerical integration methods in celestial mechanics. It is found that the Lyapunov Characteristic Numbers are extremely useful for the classification of the solutions of nonintegrable dynamical systems, especially in order to distinguish between quasi-periodic and chaotic solutions. However, the Lyapunov Characteristics Numbers do not appear to be useful for the purpose of evaluating numerical integration methods.
The Lyapunov stabilization of satellite equations of motion using integrals
Nacozy, P. E.
1973-01-01
A method is introduced that weakens the Lyapunov or in track instability of satellite equations of motion. The method utilizes a linearized energy integral of satellite motion as a constraint on solutions obtained by numerical integration. The procedure prevents local numerical error from altering the frequency associated with the fast angular variable and thereby reduces the Lyapunov instability and the global numerical error. Applications of the method to satellite motion show accuracy improvements of two to three orders of magnitude in position and velocity after 50 revolutions. A modification of the method is presented that allows the use of slowly varying integrals of motion.
The Lyapunov stabilization of satellite equations of motion using integrals
Nacozy, P. E.
1973-01-01
A method is introduced that weakens the Lyapunov or in track instability of satellite equations of motion. The method utilizes a linearized energy integral of satellite motion as a constraint on solutions obtained by numerical integration. The procedure prevents local numerical error from altering the frequency associated with the fast angular variable and thereby reduces the Lyapunov instability and the global numerical error. Applications of the method to satellite motion show accuracy improvements of two to three orders of magnitude in position and velocity after 50 revolutions. A modification of the method is presented that allows the use of slowly varying integrals of motion.
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
The asymptotic Lyapunov stability of one quasi-integrable Hamiltonian system with time-delayed feedback control is studied by using Lyapunov functions and stochastic averaging method.First,a quasi-integrable Hamiltonian system with time-delayed feedback control subjected to Gaussian white noise excitation is approximated by a quasi-integrable Hamiltonian system without time delay.Then,stochastic averaging method for quasi-integrable Hamiltonian system is used to reduce the dimension of the original system,and after that the Lyapunov function of the averaged It? equation is taken as the optimal linear combination of the corresponding independent first integrals in involution.Finally,the stability of the system is determined by using the largest eigenvalue of the linearized system.Two examples are used to illustrate the proposed procedure and the effects of delayed time on the Lyapunov stability are discussed as well.
Stability of dynamical systems on the role of monotonic and non-monotonic Lyapunov functions
Michel, Anthony N; Liu, Derong
2015-01-01
The second edition of this textbook provides a single source for the analysis of system models represented by continuous-time and discrete-time, finite-dimensional and infinite-dimensional, and continuous and discontinuous dynamical systems. For these system models, it presents results which comprise the classical Lyapunov stability theory involving monotonic Lyapunov functions, as well as corresponding contemporary stability results involving non-monotonicLyapunov functions.Specific examples from several diverse areas are given to demonstrate the applicability of the developed theory to many important classes of systems, including digital control systems, nonlinear regulator systems, pulse-width-modulated feedback control systems, and artificial neural networks. The authors cover the following four general topics: - Representation and modeling of dynamical systems of the types described above - Presentation of Lyapunov and Lagrange stability theory for dynamical sy...
The Boundary Crossing Theorem and the Maximal Stability Interval
Directory of Open Access Journals (Sweden)
Jorge-Antonio López-Renteria
2011-01-01
useful tools in the study of the stability of family of polynomials. Although both of these theorem seem intuitively obvious, they can be used for proving important results. In this paper, we give generalizations of these two theorems and we apply such generalizations for finding the maximal stability interval.
Blackwell, C. C.
1987-01-01
A relevant facet of the application of Lyapunov gradient-generated robust control to unstable linear autonomous plants is explored. It is demonstrated that if the plant, the output, and the nominal stabilizing control satisfy certain conditions, then the robust component alone stabilizes the nominal plant. An example characterized by two zero eigenvalues and two negative real value poles is presented. These results assure that the robust component will fulfill the role of nominal stabilization successfully so long as the possible magnitude of the robust component can overcome the contribution of the instability to positiveness of the Lyapunov rate.
Blackwell, C. C.
1987-01-01
A relevant facet of the application of Lyapunov gradient-generated robust control to unstable linear autonomous plants is explored. It is demonstrated that if the plant, the output, and the nominal stabilizing control satisfy certain conditions, then the robust component alone stabilizes the nominal plant. An example characterized by two zero eigenvalues and two negative real value poles is presented. These results assure that the robust component will fulfill the role of nominal stabilization successfully so long as the possible magnitude of the robust component can overcome the contribution of the instability to positiveness of the Lyapunov rate.
Energy Technology Data Exchange (ETDEWEB)
Blanchini, F. [Universita di Udine (Italy); Carabelli, S. [Politecnico di Torino (Italy)
1994-12-31
We apply a technique recently proposed in literature for the robust stabilization of linear systems with time-varying uncertain parameters to a magnetic levitation system. This technique allows the construction of a polyhedral Lyapunov function and a linear variable-structure stabilizing controller.
A Lyapunov-Krasovskii methodology for asymptotic stability of discrete time delay systems
Directory of Open Access Journals (Sweden)
Stojanović Sreten B.
2007-01-01
Full Text Available This paper presents a Lyapunov-Krasovskii methodology for asymptotic stability of discrete time delay systems. Based on the methods, delay-independent stability condition is derived. A numerical example has been working out to show the applicability of results derived.
Stabilization of Parametric Roll Resonance with Active U-Tanks via Lyapunov Control Design
DEFF Research Database (Denmark)
Holden, Christian; Galeazzi, Roberto; Fossen, Thor Inge;
2009-01-01
design of an active u-tank stabilizer is carried out using Lyapunov theory. A nonlinear backstepping controller is developed to provide global exponential stability of roll. An extension of commonly used u-tank models is presented to account for large roll angles, and the control design is tested via...
Directory of Open Access Journals (Sweden)
Héctor Armando Durán Peralta
2010-04-01
Full Text Available The stability of reactors having encompassing concentration and temperature parameters, such as continuous flow stirred tank reactors (CSTR, has been widely explored in the literature; however, there are few papers about the stability of tubular reactor having distributed spatial concentration and temperature parameters such as the plow flow tubular reactor (PFTR. This paper analyses the stability of isothermal and non-isothermal PFTR reactors using the Lyapunov functional method. The first order kinetic reaction was selected because one of this paper’s oblectives was to apply Lyapunov functionals to stability analysis of distributed parameter reactors (technique used in electrical engineering systems’ stability analysis. The stability analysis revealed asymptotically stable tempe- rature and concentration profiles for isothermal PFTR, non-isothermal PFTR with kinetic constant independent of temperature and adiabatic non-isothermal PFTR. Analysis revealed an asymptotically stability region for the heat exchange reactor and an uncertain region where it may have oscillations.
Robust Stabilization for Uncertain Control Systems Using Piecewise Quadratic Lyapunov Functions
Institute of Scientific and Technical Information of China (English)
无
2002-01-01
The sufficient condition based on piecewise quadratic simultaneous Lyapunov functions for robust stabilizationof uncertain control systems via a constant linear state feedback control law is obtained. The objective is to use a robuststability criterion that is less conservative than the usual quadratic stability criterion. Numerical example is given, show-ing the advanteges of the proposed method.
Stability of stationary barotropic modons by Lyapunov's direct method
Sakuma, H.; Ghil, M.
1990-01-01
A new Liapunov stability condition is formulated for the shallow-water equations, using a gage-variable formalism. This sufficient condition is derived for the class of perturbations that conserve the total mass. It is weaker than existing stability criteria, i.e., it applies to a wider class of flows. Formal stability to infinitesimally small perturbations of arbitrary shape is obtained for two classes of large-scale geophysical flows: pseudo-eastward flow with constant shear, and localized coherent structures of modon type.
Stability analysis for impulsive fractional hybrid systems via variational Lyapunov method
Yang, Ying; He, Yong; Wang, Yong; Wu, Min
2017-04-01
This paper investigates the stability properties for a class of impulsive Caputo fractional-order hybrid systems with impulse effects at fixed moments. By utilizing the variational Lyapunov method, a fractional variational comparison principle is established. Some stability and instability criteria in terms of two measures are obtained. These results generalize the known ones, extending the corresponding theory of impulsive fractional differential systems. An example is given to demonstrate their effectiveness.
Stability of synchronization in coupled time-delay systems using Krasovskii-Lyapunov theory.
Senthilkumar, D V; Kurths, J; Lakshmanan, M
2009-06-01
Stability of synchronization in unidirectionally coupled time-delay systems is studied using the Krasovskii-Lyapunov theory. We have shown that the same general stability condition is valid for different cases, even for the general situation (but with a constraint) where all the coefficients of the error equation corresponding to the synchronization manifold are time dependent. These analytical results are also confirmed by the numerical simulation of paradigmatic examples.
H2-Stabilization of the Isothermal Euler Equations:a Lyapunov Function Approach
Institute of Scientific and Technical Information of China (English)
Martin GUGAT; Günter LEUGERING; Simona TAMASOIU; Ke WANG
2012-01-01
The authors consider the problem of boundary feedback stabilization of the 1D Euler gas dynamics locally around stationary states and prove the exponential stability with respect to the H2-norm.To this end,an explicit Lyapunov function as a weighted and squared H2-norm of a small perturbation of the stationary solution is constructed.The authors show that by a suitable choice of the boundary feedback conditions,the H2-exponential stability of the stationary solution follows.Due to this fact,the system is stabilized over an infinite time interval.Furthermore,exponential estimates for the C1-norm are derived.
Chu, Chia-Chi; Tsai, Hung-Chi; Chang, Wei-Neng
A Lyapunov-based recurrent neural networks unified power flow controller (UPFC) is developed for improving transient stability of power systems. First, a simple UPFC dynamical model, composed of a controllable shunt susceptance on the shunt side and an ideal complex transformer on the series side, is utilized to analyze UPFC dynamical characteristics. Secondly, we study the control configuration of the UPFC with two major blocks: the primary control, and the supplementary control. The primary control is implemented by standard PI techniques when the power system is operated in a normal condition. The supplementary control will be effective only when the power system is subjected by large disturbances. We propose a new Lyapunov-based UPFC controller of the classical single-machine-infinite-bus system for damping enhancement. In order to consider more complicated detailed generator models, we also propose a Lyapunov-based adaptive recurrent neural network controller to deal with such model uncertainties. This controller can be treated as neural network approximations of Lyapunov control actions. In addition, this controller also provides online learning ability to adjust the corresponding weights with the back propagation algorithm built in the hidden layer. The proposed control scheme has been tested on two simple power systems. Simulation results demonstrate that the proposed control strategy is very effective for suppressing power swing even under severe system conditions.
On the asymptotic stability of linear discrete time-delay systems: The Lyapunov approach
Directory of Open Access Journals (Sweden)
Stojanović Sreten B.
2006-01-01
Full Text Available New conditions for the stability of discrete delay systems of the form x (k+1 = Arjx (k + Aix (k-h are presented in the paper. These new delay-independent conditions were derived using an approach based on the second Lyapunov's method. These results are less conservative than some in the existing literature. A numerical example was worked out to show the applicability of the derived results.
Lyapunov stability in an evolutionary game theory model of the labour market
Directory of Open Access Journals (Sweden)
Ricardo Azevedo Araujo
2014-01-01
Full Text Available In this paper the existence and stability of equilibriums in an evolutionary game theory model of the labour market is studied by using the Lyapunov method. The model displays multiple equilibriums and it is shown that the Nash equilibriums of the static game are evolutionary stable equilibrium in the game theory evolutionary set up. A complete characterization of the dynamics of an evolutionary model of the labour market is provided.
Lyapunov stability and its application to systems of ordinary differential equations
Kennedy, E. W.
1979-01-01
An outline and a brief introduction to some of the concepts and implications of Lyapunov stability theory are presented. Various aspects of the theory are illustrated by the inclusion of eight examples, including the Cartesian coordinate equations of the two-body problem, linear and nonlinear (Van der Pol's equation) oscillatory systems, and the linearized Kustaanheimo-Stiefel element equations for the unperturbed two-body problem.
Razumikhin Stability Theorem for Fractional Systems with Delay
Directory of Open Access Journals (Sweden)
D. Baleanu
2010-01-01
Full Text Available Fractional calculus techniques and methods started to be applied successfully during the last decades in several fields of science and engineering. In this paper we studied the stability of fractional-order nonlinear time-delay systems for Riemann-Liouville and Caputo derivatives and we extended Razumikhin theorem for the fractional nonlinear time-delay systems.
Zhang, Baoyong; Lam, James; Xu, Shengyuan
2015-07-01
This paper revisits the problem of asymptotic stability analysis for neural networks with distributed delays. The distributed delays are assumed to be constant and prescribed. Since a positive-definite quadratic functional does not necessarily require all the involved symmetric matrices to be positive definite, it is important for constructing relaxed Lyapunov-Krasovskii functionals, which generally lead to less conservative stability criteria. Based on this fact and using two kinds of integral inequalities, a new delay-dependent condition is obtained, which ensures that the distributed delay neural network under consideration is globally asymptotically stable. This stability criterion is then improved by applying the delay partitioning technique. Two numerical examples are provided to demonstrate the advantage of the presented stability criteria.
Relative Lyapunov Center Bifurcations
DEFF Research Database (Denmark)
Wulff, Claudia; Schilder, Frank
2014-01-01
Relative equilibria (REs) and relative periodic orbits (RPOs) are ubiquitous in symmetric Hamiltonian systems and occur, for example, in celestial mechanics, molecular dynamics, and rigid body motion. REs are equilibria, and RPOs are periodic orbits of the symmetry reduced system. Relative Lyapunov...... center bifurcations are bifurcations of RPOs from REs corresponding to Lyapunov center bifurcations of the symmetry reduced dynamics. In this paper we first prove a relative Lyapunov center theorem by combining recent results on the persistence of RPOs in Hamiltonian systems with a symmetric Lyapunov...... center theorem of Montaldi, Roberts, and Stewart. We then develop numerical methods for the detection of relative Lyapunov center bifurcations along branches of RPOs and for their computation. We apply our methods to Lagrangian REs of the N-body problem....
Colburn, B. K.; Boland, J. S., III
1976-01-01
A new nonlinear stability criterion is developed by use of a class of Lyapunov functionals for model-reference adaptive systems (MRAS). Results are compared with traditional results, and a comparative design technique is used to illustrate its function in improving the transient response of an MRAS controller. For a particular system structure and class of input signals, the new stability criterion is shown to include traditional sufficiency stability conditions as a special case. An example is cited to illustrate the use of the nonlinear criterion and its definite advantages in helping improve the adaptive error transient response of a system. Analysis of results is effected by use of a linearization technique on the resulting adaptive equations.
Colburn, B. K.; Boland, J. S., III
1976-01-01
A new nonlinear stability criterion is developed by use of a class of Lyapunov functionals for model-reference adaptive systems (MRAS). Results are compared with traditional results, and a comparative design technique is used to illustrate its function in improving the transient response of an MRAS controller. For a particular system structure and class of input signals, the new stability criterion is shown to include traditional sufficiency stability conditions as a special case. An example is cited to illustrate the use of the nonlinear criterion and its definite advantages in helping improve the adaptive error transient response of a system. Analysis of results is effected by use of a linearization technique on the resulting adaptive equations.
A Lyapunov method for stability analysis of piecewise-affine systems over non-invariant domains
Rubagotti, Matteo; Zaccarian, Luca; Bemporad, Alberto
2016-05-01
This paper analyses stability of discrete-time piecewise-affine systems, defined on possibly non-invariant domains, taking into account the possible presence of multiple dynamics in each of the polytopic regions of the system. An algorithm based on linear programming is proposed, in order to prove exponential stability of the origin and to find a positively invariant estimate of its region of attraction. The results are based on the definition of a piecewise-affine Lyapunov function, which is in general discontinuous on the boundaries of the regions. The proposed method is proven to lead to feasible solutions in a broader range of cases as compared to a previously proposed approach. Two numerical examples are shown, among which a case where the proposed method is applied to a closed-loop system, to which model predictive control was applied without a-priori guarantee of stability.
Fixed point theorems and stability of iterations in cone metric spaces
Directory of Open Access Journals (Sweden)
Yuan Qing
2012-04-01
Full Text Available In this paper, fixed point problems of weak contractions are investigated in cone metric spaces. Theorems of convergence and theorems of stability for fixed points of some weak contraction are established in cone metric spaces.
Sun, Yuming; Wu, Christine Qiong
2012-12-01
Balancing control is important for biped standing. In spite of large efforts, it is very difficult to design balancing control strategies satisfying three requirements simultaneously: maintaining postural stability, improving energy efficiency and satisfying the constraints between the biped feet and the ground. In this article, a proportional-derivative (PD) controller is proposed for a standing biped, which is simplified as a two-link inverted pendulum with one additional rigid foot-link. The genetic algorithm (GA) is used to search for the control gain meeting all three requirements. The stability analysis of such a deterministic biped control system is carried out using the concept of Lyapunov exponents (LEs), based on which, the system stability, where the disturbance comes from the initial states, and the structural stability, where the disturbance comes from the PD gains, are examined quantitively in terms of stability region. This article contributes to the biped balancing control, more significantly, the method shown in the studied case of biped provides a general framework of systematic stability analysis for certain deterministic nonlinear dynamical systems.
Robust Stability Analysis of Nonlinear Switched Systems with Filippov Solutions
DEFF Research Database (Denmark)
Ahmadi, Mohamadreza; Mojallali, Hamed; Wisniewski, Rafal
2012-01-01
. Based on the theory of differential inclusions, a Lyapunov stability theorem is brought forward. These results are also extended to autonomous switched systems subject to polytopic uncertainty. Furthermore, the proposed stability theorems are reformulated using the sum of squares decomposition method...... which provides sufficient means to construct the corresponding Lyapunov functions via available semi-definite programming techniques....
Zhang, Hongbin; Feng, Gang
2008-10-01
This paper is concerned with stability analysis and H(infinity) decentralized control of discrete-time fuzzy large-scale systems based on piecewise Lyapunov functions. The fuzzy large-scale systems consist of J interconnected discrete-time Takagi-Sugeno (T-S) fuzzy subsystems, and the stability analysis is based on Lyapunov functions that are piecewise quadratic. It is shown that the stability of the discrete-time fuzzy large-scale systems can be established if a piecewise quadratic Lyapunov function can be constructed, and moreover, the function can be obtained by solving a set of linear matrix inequalities (LMIs) that are numerically feasible. The H(infinity) controllers are also designed by solving a set of LMIs based on these powerful piecewise quadratic Lyapunov functions. It is demonstrated via numerical examples that the stability and controller synthesis results based on the piecewise quadratic Lyapunov functions are less conservative than those based on the common quadratic Lyapunov functions.
W-Stability of Multistable Nonlinear Discrete-Time Systems
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Zhishuai Ding
2012-01-01
Full Text Available Motivated by the importance and application of discrete dynamical systems, this paper presents a new Lyapunov characterization which is an extension of conventional Lyapunov characterization for multistable discrete-time nonlinear systems. Based on a new type stability notion of W-stability introduced by D. Efimov, the estimates of solution and the Lyapunov stability theorem and converse theorem are proposed for multi-stable discrete-time nonlinear systems.
A Lyapunov Stability Theory-Based Control Strategy for Three-Level Shunt Active Power Filter
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Yijia Cao
2017-01-01
Full Text Available The three-phase three-wire neutral-point-clamped shunt active power filter (NPC-SAPF, which most adopts classical closed-loop feedback control methods such as proportional-integral (PI, proportional-resonant (PR and repetitive control, can only output 1st–25th harmonic currents with 10–20 kHz switching frequency. The reason for this is that the controller design must make a compromise between system stability and harmonic current compensation ability under the condition of less than 20 kHz switching frequency. To broaden the bandwidth of the compensation current, a Lyapunov stability theory-based control strategy is presented in this paper for NPC-SAPF. The proposed control law is obtained by constructing the switching function on the basis of the mathematical model and the Lyapunov candidate function, which can avoid introducing closed-loop feedback control and keep the system globally asymptotically stable. By means of the proposed method, the NPC-SAPF has compensation ability for the 1st–50th harmonic currents, the total harmonic distortion (THD and each harmonic content of grid currents satisfy the requirements of IEEE Standard 519-2014. In order to verify the superiority of the proposed control strategy, stability conditions of the proposed strategy and the representative PR controllers are compared. The simulation results in MATLAB/Simulink (MathWorks, Natick, MA, USA and the experimental results obtained on a 6.6 kVA NPC-SAPF laboratory prototype validate the proposed control strategy.
Jupiter's winds and Arnol'd's second stability theorem: Slowly moving waves and neutral stability
Stamp, Andrew P.; Dowling, Timothy E.
1993-01-01
Since the Voyager encounters in 1979, it has been known that Jupiter's cloud-top zonal winds violate the barotropic stability criterion. A vortex-tube stretching analysis of the Voyager wind data indicates that the more general Charney-Stern stability criterion is also violated. On the other hand, the zonal winds determined by tracking cloud features in Hubble Space Telescope images taken in 1991 precisely match the zonal winds determined by tracking cloud features in Voyager images, and it is hard to understand how a complicated zonal wind profile like Jupiter's could be unstable and yet not change at all in 12 years. In fact, there are at least two unknown ways to violate the Charney-Stern stability criterion and still have a stable flow. The better known of these is called Fjortoft's theorem, or Arnol'd's 1st theorem for the case of large-amplitude perturbations. Although the Fjortoft-Arnol'd theorem has been extended from the quasi-geostrophic equations to the primitive equations, the basic requirement that the potential vorticity be an increasing function of streamfunction is opposite to the case found in Jupiter, where the Voyager data indicate that the potential vorticity is a decreasing function of streamfunction. But this second case is precisely that which is covered by Arnol'd's 2nd stability theorem. In fact, the Voyager data suggest that Jupiter's zonal winds are neutrally stable with respect to Arnol'd's 2nd stability theorem. Here, we analyze the linear stability problem of a one-parameter family of sinusoidal zonal wind profiles that are close to neutral stability with respect to Arnol'd's 2nd stability theorem. We find numerically that the most unstable mode is always stationary, which may help to explain the slowly moving mode 10 waves observed on Jupiter. We find that violation of Arnol'd's 2nd stability theorem is both necessary and sufficient for instability of sinusoidal profiles. However, there appears to be no simple extension of Arnol'd's 2
Energy Technology Data Exchange (ETDEWEB)
Look, Nicole [Department of Applied Mathematics, University of Colorado Boulder, Boulder, Colorado 80309 (United States); Arellano, Christopher J.; Grabowski, Alena M.; Kram, Rodger [Department of Integrative Physiology, University of Colorado Boulder, Boulder, Colorado 80309 (United States); McDermott, William J. [The Orthopedic Specialty Hospital, Murray, Utah 84107 (United States); Bradley, Elizabeth [Department of Computer Science, University of Colorado Boulder, Boulder, Colorado 80309, USA and Santa Fe Institute, Santa Fe, New Mexico 87501 (United States)
2013-12-15
In this paper, we study dynamic stability during running, focusing on the effects of speed, and the use of a leg prosthesis. We compute and compare the maximal Lyapunov exponents of kinematic time-series data from subjects with and without unilateral transtibial amputations running at a wide range of speeds. We find that the dynamics of the affected leg with the running-specific prosthesis are less stable than the dynamics of the unaffected leg and also less stable than the biological legs of the non-amputee runners. Surprisingly, we find that the center-of-mass dynamics of runners with two intact biological legs are slightly less stable than those of runners with amputations. Our results suggest that while leg asymmetries may be associated with instability, runners may compensate for this effect by increased control of their center-of-mass dynamics.
A Lyapunov-Razumikhin approach for stability analysis of logistics networks with time-delays
Dashkovskiy, Sergey; Karimi, Hamid Reza; Kosmykov, Michael
2012-05-01
Logistics network represents a complex system where different elements that are logistic locations interact with each other. This interaction contains delays caused by time needed for delivery of the material. Complexity of the system, time-delays and perturbations in a customer demand may cause unstable behaviour of the network. This leads to the loss of the customers and high inventory costs. Thus the investigation of the network on stability is desired during its design. In this article we consider local input-to-state stability of such logistics networks. Their behaviour is described by a functional differential equation with a constant time-delay. We are looking for verifiable conditions that guarantee stability of the network under consideration. Lyapunov-Razumikhin functions and the local small gain condition are utilised to obtain such conditions. Our stability conditions for the logistics network are based on the information about the interconnection properties between logistic locations and their production rates. Finally, numerical results are provided to demonstrate the proposed approach.
Design and Lyapunov Stability Analysis of a Fuzzy Logic Controller for Autonomous Road Following
Directory of Open Access Journals (Sweden)
Yi Fu
2010-01-01
Full Text Available Autonomous road following is one of the major goals in intelligent vehicle applications. The development of an autonomous road following embedded system for intelligent vehicles is the focus of this paper. A fuzzy logic controller (FLC is designed for vision-based autonomous road following. The stability analysis of this control system is addressed. Lyapunov's direct method is utilized to formulate a class of control laws that guarantee the convergence of the steering error. Certain requirements for the control laws are presented for designers to choose a suitable rule base for the fuzzy controller in order to make the system stable. Stability of the proposed fuzzy controller is guaranteed theoretically and also demonstrated by simulation studies and experiments. Simulations using the model of the four degree of freedom nonholonomic robotic vehicle are conducted to investigate the performance of the fuzzy controller. The proposed fuzzy controller can achieve the desired steering angle and make the robotic vehicle follow the road successfully. Experiments show that the developed intelligent vehicle is able to follow a mocked road autonomously.
Kumar, Rajesh; Srivastava, Smriti; Gupta, J R P
2017-03-01
In this paper adaptive control of nonlinear dynamical systems using diagonal recurrent neural network (DRNN) is proposed. The structure of DRNN is a modification of fully connected recurrent neural network (FCRNN). Presence of self-recurrent neurons in the hidden layer of DRNN gives it an ability to capture the dynamic behaviour of the nonlinear plant under consideration (to be controlled). To ensure stability, update rules are developed using lyapunov stability criterion. These rules are then used for adjusting the various parameters of DRNN. The responses of plants obtained with DRNN are compared with those obtained when multi-layer feed forward neural network (MLFFNN) is used as a controller. Also, in example 4, FCRNN is also investigated and compared with DRNN and MLFFNN. Robustness of the proposed control scheme is also tested against parameter variations and disturbance signals. Four simulation examples including one-link robotic manipulator and inverted pendulum are considered on which the proposed controller is applied. The results so obtained show the superiority of DRNN over MLFFNN as a controller. Copyright © 2017 ISA. Published by Elsevier Ltd. All rights reserved.
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M Seidi
2016-12-01
Full Text Available Lyapunov exponent method is one of the best tools for investigating the range of stability and the transient behavior of the dynamical systems. In beryllium-moderated and heavy water-moderated reactors, photo-neutron plays an important role in dynamic behavior of the reactor. Therefore, stability analysis for changes in the control parameters of the reactor in order to guarantee safety and control nuclear reactor is important. In this work, the range of stability has been investigated using Lyapunov exponent method in response to step, ramp and sinusoidal external reactivities regarding six groups of delayed neutrons plus nine groups of photo-neutrons. The qualitative results are in good agreement with quantitative results of other works
Stability of Cellular Automata Trajectories Revisited: Branching Walks and Lyapunov Profiles
Baetens, Jan M.; Gravner, Janko
2016-10-01
We study non-equilibrium defect accumulation dynamics on a cellular automaton trajectory: a branching walk process in which a defect creates a successor on any neighborhood site whose update it affects. On an infinite lattice, defects accumulate at different exponential rates in different directions, giving rise to the Lyapunov profile. This profile quantifies instability of a cellular automaton evolution and is connected to the theory of large deviations. We rigorously and empirically study Lyapunov profiles generated from random initial states. We also introduce explicit and computationally feasible variational methods to compute the Lyapunov profiles for periodic configurations, thus developing an analog of Floquet theory for cellular automata.
On formalism and stability of switched systems
DEFF Research Database (Denmark)
Leth, John-Josef; Wisniewski, Rafal
2012-01-01
In this paper, we formulate a uniform mathematical framework for studying switched systems with piecewise linear partitioned state space and state dependent switching. Based on known results from the theory of differential inclusions, we devise a Lyapunov stability theorem suitable for this class...... of switched systems. With this, we prove a Lyapunov stability theorem for piecewise linear switched systems by means of a concrete class of Lyapunov functions. Contrary to existing results on the subject, the stability theorems in this paper include Filippov (or relaxed) solutions and allow infinite switching...
An Integral Mean Value Theorem concerning Two Continuous Functions and Its Stability
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Monea Mihai
2015-01-01
Full Text Available The aim of this paper is to investigate an integral mean value theorem proposed by one of the references of this paper. Unfortunately, the proof contains a gap. First, we present a counterexample which shows that this theorem fails in this form. Then, we present two improved versions of this theorem. The stability of the mean point arising from the second result concludes this paper.
Integral expressions of Lyapunov exponents for autonomous ordinary differential systems
Institute of Scientific and Technical Information of China (English)
DAI XiongPing
2009-01-01
In the paper,the author addresses the Lyapunov characteristic spectrum of an ergodic autonomous ordinary differential system on a complete riemannian manifold of finite dimension such as the d-dimensional euclidean space Rd,not necessarily compact,by Liaowise spectral theorems that give integral expressions of Lyapunov exponents.In the context of smooth linear skew-product flows with Polish driving systems,the results are still valid.This paper seems to be an interesting contribution to the stability theory of ordinary differential systems with non-compact phase spaces.
Integral expressions of Lyapunov exponents for autonomous ordinary differential systems
Institute of Scientific and Technical Information of China (English)
无
2009-01-01
In the paper, the author addresses the Lyapunov characteristic spectrum of an ergodic autonomous ordinary differential system on a complete riemannian manifold of finite dimension such as the d-dimensional euclidean space Rd, not necessarily compact, by Liaowise spectral theorems that give integral expressions of Lyapunov exponents. In the context of smooth linear skew-product flows with Polish driving systems, the results are still valid. This paper seems to be an interesting contribution to the stability theory of ordinary differential systems with non-compact phase spaces.
Large-Signal Lyapunov-Based Stability Analysis of DC/AC Inverters and Inverter-Based Microgrids
Kabalan, Mahmoud
Microgrid stability studies have been largely based on small-signal linearization techniques. However, the validity and magnitude of the linearization domain is limited to small perturbations. Thus, there is a need to examine microgrids with large-signal nonlinear techniques to fully understand and examine their stability. Large-signal stability analysis can be accomplished by Lyapunov-based mathematical methods. These Lyapunov methods estimate the domain of asymptotic stability of the studied system. A survey of Lyapunov-based large-signal stability studies showed that few large-signal studies have been completed on either individual systems (dc/ac inverters, dc/dc rectifiers, etc.) or microgrids. The research presented in this thesis addresses the large-signal stability of droop-controlled dc/ac inverters and inverter-based microgrids. Dc/ac power electronic inverters allow microgrids to be technically feasible. Thus, as a prelude to examining the stability of microgrids, the research presented in Chapter 3 analyzes the stability of inverters. First, the 13 th order large-signal nonlinear model of a droop-controlled dc/ac inverter connected to an infinite bus is presented. The singular perturbation method is used to decompose the nonlinear model into 11th, 9th, 7th, 5th, 3rd and 1st order models. Each model ignores certain control or structural components of the full order model. The aim of the study is to understand the accuracy and validity of the reduced order models in replicating the performance of the full order nonlinear model. The performance of each model is studied in three different areas: time domain simulations, Lyapunov's indirect method and domain of attraction estimation. The work aims to present the best model to use in each of the three domains of study. Results show that certain reduced order models are capable of accurately reproducing the performance of the full order model while others can be used to gain insights into those three areas of
Lyapunov vs. geometrical stability analysis of the Kepler and the restricted three body problems
DEFF Research Database (Denmark)
Yahalom, A.; Levitan, J.; Lewkowicz, M.
2011-01-01
to move in a very interesting and intricate but periodic trajectory; however, the standard Lyapunov analysis, as well as methods based on the parametric variation of curvature associated with the Jacobi metric, incorrectly predict chaotic behavior. The geometric approach predicts the correct stable motion...
Stability of quantized time-delay nonlinear systems : A Lyapunov-Krasowskii-functional approach
Persis, Claudio De; Mazenc, Frédéric
2009-01-01
Lyapunov-Krasowskii functionals are used to design quantized control laws for nonlinear continuous-time systems in the presence of time-invariant constant delays in the input. The quantized control law is implemented via hysteresis to avoid chattering. Under appropriate conditions, our analysis appl
Stability analysis and quasinormal modes of Reissner–Nordstrøm space-time via Lyapunov exponent
Indian Academy of Sciences (India)
PRADHAN PARTHAPRATIM
2016-07-01
We explicitly derive the proper-time (τ ) principal Lyapunov exponent (λp) and coordinate-time (t ) principal Lyapunov exponent $(\\lambda_c)$ for Reissner–Nordstrøm (RN) black hole (BH). We also compute their ratio. For RN space-time, it is shown that the ratio is $(\\lambda_{p}/\\lambda_{c}) = r_{0}/\\sqrt{r^{2}0 − 3Mr_{0} + 2Q^{2}}$ for time-like circulargeodesics and for Schwarzschild BH, it is $(\\lambda_{p}/\\lambda_{c}) = \\sqrt{r_{0}}/\\sqrt{r_{0} − 3M}. We further show that their ratio $\\lambda_{p}/\\lambda_{c}$ may vary from orbit to orbit. For instance, for Schwarzschild BH at the innermost stable circular orbit (ISCO), the ratio is $(\\lambda_{p}/\\lambda_{c})_{|rISCO}=6M = \\sqrt{2}$ and at marginally bound circular orbit (MBCO) the ratio is calculated to be $(\\lambda_{p}/\\lambda_{c})|_{rmb}=4M = 2$. Similarly, for extremal RN BH, the ratio at ISCO is $(\\lambda_{p}/\\lambda_{c})|_{rISCO}=4M = 2\\sqrt{2}/\\sqrt{3}$. We also further analyse the geodesic stability via this exponent. By evaluating the Lyapunov exponent, it is shown that in the eikonal limit, the real and imaginary parts of the quasinormal modes of RN BH is given by the frequency and instability time-scale of the unstable null circular geodesics.
EVENTUAL STABILITY OF IMPULSIVE DIFFERENTIAL SYSTEMS
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
In this article, criteria of eventual stability are established for impulsive differential systems using piecewise continuous Lyapunov functions. The sufficient conditions that are obtained significantly depend on the moments of impulses. An example is discussed to illustrate the theorem.
Using largest Lyapunov exponent to confirm the intrinsic stability of boiling water reactors
Energy Technology Data Exchange (ETDEWEB)
Gavilian-Moreno, Carlos [Iberdrola Generacion, S.A., Cofrentes Nuclear Power Plant, Project Engineering Department, Paraje le Plano S/N, Valencia (Spain); Espinosa-Paredes, Gilberto [Area de ingeniera en Recursos Energeticos, Universidad Autonoma Metropolitana-Iztapalapa, Mexico city (Mexico)
2016-04-15
The aim of this paper is the study of instability state of boiling water reactors with a method based in largest Lyapunov exponents (LLEs). Detecting the presence of chaos in a dynamical system is an important problem that is solved by measuring the LLE. Lyapunov exponents quantify the exponential divergence of initially close state-space trajectories and estimate the amount of chaos in a system. This method was applied to a set of signals from several nuclear power plant (NPP) reactors under commercial operating conditions that experienced instabilities events, apparently each of a different nature. Laguna Verde and Forsmark NPPs with in-phase instabilities, and Cofrentes NPP with out-of-phases instability. This study presents the results of intrinsic instability in the boiling water reactors of three NPPs. In the analyzed cases the limit cycle was not reached, which implies that the point of equilibrium exerts influence and attraction on system evolution.
The nekhoroshev theorem and long-term stabilities in the solar system
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Guzzo M.
2015-01-01
Full Text Available The Nekhoroshev theorem has been often indicated in the last decades as the reference theorem for explaining the dynamics of several systems which are stable in the long-term. The Solar System dynamics provides a wide range of possible and useful applications. In fact, despite the complicated models which are used to numerically integrate realistic Solar System dynamics as accurately as possible, when the integrated solutions are chaotic the reliability of the numerical integrations is limited, and a theoretical long-term stability analysis is required. After the first formulation of Nekhoroshev’s theorem in 1977, many theoretical improvements have been achieved. On the one hand, alternative proofs of the theorem itself led to consistent improvements of the stability estimates; on the other hand, the extensions which were necessary to apply the theorem to the systems of interest for Solar System Dynamics, in particular concerning the removal of degeneracies and the implementation of computer assisted proofs, have been developed. In this review paper we discuss some of the motivations and the results which have made Nekhoroshev’s theorem a reference stability result for many applications in the Solar System dynamics.
Azizi, Sajad
2017-05-01
The robust stability of a class of feedback linearizable minimum-phase nonlinear system, having parametric uncertainties, is investigated in this study. The system in new coordinates is represented to an equivalent formulation after the attempt of feedback linearization. Due to the parametric uncertainties the approximately linearized system entails a norm bounded input nonlinearity such that the equilibrium point condition in error dynamics can not be satisfied. Accordingly, to guarantee the regional asymptotic stability a control synthesis problem is proposed by means of sufficient Linear Matrix Inequalities (LMIs) together with an amended nonlinear control term, derived from the Lyapunov redesign method, which tackles zero steady-state error condition. The numerical examples of a general aviation aircraft's longitudinal dynamics and inverted pendulum are simulated to show the proficiency of the proposed control technique. Copyright © 2017 ISA. Published by Elsevier Ltd. All rights reserved.
LAGRANGE STABILITY IN MEAN SQUARE OF STOCHASTIC REACTION DIFFUSION EQUATIONS
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
This work is devoted to the discussion of stochastic reaction diffusion equations and some new theorems on Lagrange stability in mean square of the solution are established via Lyapunov method which is nothing to be done in the past.
Conditions of asymptotic stability for linear homogeneous switched systems
Ivanov, Gennady; Alferov, Gennady; Sharlay, Artem; Efimova, Polina
2017-07-01
In this article the authors prove the theorems giving the necessary and sufficient conditions for stability of robotic and mechatronic systems motion in terms of Lyapunov functions theory with the use of set-theoretic approach.
Crauel, Hans; Eckmann, Jean-Pierre
1991-01-01
Since the predecessor to this volume (LNM 1186, Eds. L. Arnold, V. Wihstutz)appeared in 1986, significant progress has been made in the theory and applications of Lyapunov exponents - one of the key concepts of dynamical systems - and in particular, pronounced shifts towards nonlinear and infinite-dimensional systems and engineering applications are observable. This volume opens with an introductory survey article (Arnold/Crauel) followed by 26 original (fully refereed) research papers, some of which have in part survey character. From the Contents: L. Arnold, H. Crauel: Random Dynamical Systems.- I.Ya. Goldscheid: Lyapunov exponents and asymptotic behaviour of the product of random matrices.- Y. Peres: Analytic dependence of Lyapunov exponents on transition probabilities.- O. Knill: The upper Lyapunov exponent of Sl (2, R) cocycles:Discontinuity and the problem of positivity.- Yu.D. Latushkin, A.M. Stepin: Linear skew-product flows and semigroups of weighted composition operators.- P. Baxendale: Invariant me...
A STABILITY THEOREM FOR CONSTRAINED OPTIMAL CONTROL PROBLEMS
Institute of Scientific and Technical Information of China (English)
M.H. Farag
2004-01-01
This paper presents the stability of difference approximations of an optimal control problem for a quasilinear parabolic equation with controls in the coefficients, boundary conditions and additional restrictions. The optimal control problem has been convered to one of the optimization problem using a penalty function technique. The difference approximations problem for the considered problem is obtained. The estimations of stability of the solution of difference approximations problem are proved. The stability estimation of the solution of difference approximations problem by the controls is obtained.
Mittag-Leffler Stability Theorem for Fractional Nonlinear Systems with Delay
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S. J. Sadati
2010-01-01
Full Text Available Fractional calculus started to play an important role for analysis of the evolution of the nonlinear dynamical systems which are important in various branches of science and engineering. In this line of taught in this paper we studied the stability of fractional order nonlinear time-delay systems for Caputo's derivative, and we proved two theorems for Mittag-Leffler stability of the fractional nonlinear time delay systems.
Stability of Gas Clouds in Galactic Nuclei: An Extended Virial Theorem
Chen, Xian; Amaro-Seoane, Pau; Cuadra, Jorge
2016-03-01
Cold gas entering the central 1-102 pc of a galaxy fragments and condenses into clouds. The stability of the clouds determines whether they will be turned into stars or can be delivered to the central supermassive black hole (SMBH) to turn on an active galactic nucleus (AGN). The conventional criteria to assess the stability of these clouds, such as the Jeans criterion and Roche (or tidal) limit, are insufficient here, because they assume the dominance of self-gravity in binding a cloud, and neglect external agents, such as pressure and tidal forces, which are common in galactic nuclei. We formulate a new scheme for judging this stability. We first revisit the conventional Virial theorem, taking into account an external pressure, to identify the correct range of masses that lead to stable clouds. We then extend the theorem to further include an external tidal field, which is equally crucial for the stability in the region of our interest—in dense star clusters, around SMBHs. We apply our extended Virial theorem to find new solutions to controversial problems, namely, the stability of the gas clumps in AGN tori, the circum-nuclear disk in the Galactic Center, and the central molecular zone of the Milky Way. The masses we derive for these structures are orders of magnitude smaller than the commonly used Virial masses (equivalent to the Jeans mass). Moreover, we prove that these clumps are stable, contrary to what one would naively deduce from the Roche (tidal) limit.
STABILITY OF GAS CLOUDS IN GALACTIC NUCLEI: AN EXTENDED VIRIAL THEOREM
Energy Technology Data Exchange (ETDEWEB)
Chen, Xian; Cuadra, Jorge [Instituto de Astrofísica, Facultad de Física, Pontificia Universidad Católica de Chile, 782-0436 Santiago (Chile); Amaro-Seoane, Pau, E-mail: xchen@astro.puc.cl, E-mail: jcuadra@astro.puc.cl, E-mail: Pau.Amaro-Seoane@aei.mpg.de [Max Planck Institut für Gravitationsphysik (Albert-Einstein-Institut), D-14476 Potsdam (Germany)
2016-03-10
Cold gas entering the central 1–10{sup 2} pc of a galaxy fragments and condenses into clouds. The stability of the clouds determines whether they will be turned into stars or can be delivered to the central supermassive black hole (SMBH) to turn on an active galactic nucleus (AGN). The conventional criteria to assess the stability of these clouds, such as the Jeans criterion and Roche (or tidal) limit, are insufficient here, because they assume the dominance of self-gravity in binding a cloud, and neglect external agents, such as pressure and tidal forces, which are common in galactic nuclei. We formulate a new scheme for judging this stability. We first revisit the conventional Virial theorem, taking into account an external pressure, to identify the correct range of masses that lead to stable clouds. We then extend the theorem to further include an external tidal field, which is equally crucial for the stability in the region of our interest—in dense star clusters, around SMBHs. We apply our extended Virial theorem to find new solutions to controversial problems, namely, the stability of the gas clumps in AGN tori, the circum-nuclear disk in the Galactic Center, and the central molecular zone of the Milky Way. The masses we derive for these structures are orders of magnitude smaller than the commonly used Virial masses (equivalent to the Jeans mass). Moreover, we prove that these clumps are stable, contrary to what one would naively deduce from the Roche (tidal) limit.
Directory of Open Access Journals (Sweden)
Coşkun Yakar
2010-01-01
Full Text Available The qualitative behavior of a perturbed fractional-order differential equation with Caputo's derivative that differs in initial position and initial time with respect to the unperturbed fractional-order differential equation with Caputo's derivative has been investigated. We compare the classical notion of stability to the notion of initial time difference stability for fractional-order differential equations in Caputo's sense. We present a comparison result which again gives the null solution a central role in the comparison fractional-order differential equation when establishing initial time difference stability of the perturbed fractional-order differential equation with respect to the unperturbed fractional-order differential equation.
Institute of Scientific and Technical Information of China (English)
Qiu Fang; Zhang Quan-Xin; Deng Xue-Hui
2012-01-01
This paper investigates the asymptotical stability problem of a neural system with a constant delay.A new delaydependent stability condition is derived by using the novel augmented Lyapunov-Krasovskii function with triple integral terms,and the additional triple integral terms play a key role in the further reduction of conservativeness.Finally,a numerical example is given to demonstrate the effectiveness and lower conservativeness of the proposed method.
Global stabilization using LSS-Theorem: Applications to Robotics and Aerospace Vehicles
Selman, AbdulRazzak
Underactuated mechanical systems are gaining interest as they can sometimes provide the desired motion or functionality at reduced cost due to their using fewer expensive actuators. The term "underactuated" refers to the fact that such mechanical systems have fewer actuators than degrees of freedom, which makes them very difficult to control. Moreover, underactuated robots have nonlinear dynamics which must be tackled with nonlinear control techniques. Furthermore, control theory for underactuated mechanical systems has been an active area of research for the past 15-20 years. Most of the research has focused on local and global asymptotic stabilization by feedback. Underactuated systems can either possess nonminimum phase or minimum phase characteristics. For minimum phase underactuated systems, the stabilization problem is rather simple and many existing control design methodologies have been proved powerful in providing a solution to this problem. For nonminimum phase underactuated systems, asymptotic stabilization problem has been, and still is, an attractive subject to the researchers in the field of nonlinear control system and theory. In particular, global asymptotic stabilization (GAS) at a desired equilibrium point of such systems by means of a single smooth static or dynamic state feedback law is still largely an open problem in the literature. In this thesis, the problem of GAS via a smooth static state feedback law is addressed for a class of an underactuated nonlinear system that is affine (possibly non affine) in the control, partially feedback linearizable, nonminimum phase and (possibly) has a non-integrable acceleration constraint. The core result of the thesis is formulated through a theorem that the author refers to through this thesis as the Legend of Salah Salman (LSS) Theorem. LSS theorem states the existence of a smooth static state feedback law that globally asymptotically stabilizes the origin of the nonlinear underactuated system that is
Stability of Gas Clouds in Galactic Nuclei: An Extended Virial Theorem
Chen, Xian; Cuadra, Jorge
2015-01-01
Cold gas entering the central $1$ to $10^2$ pc of a galaxy fragments and condenses into clouds. The stability of the clouds determines whether they will be turned into stars or can be delivered to the central supermassive black hole (SMBH) to turn on an active galactic nucleus (AGN). The conventional criteria to assess the stability of these clouds, such as the Jeans criterion and Roche (or tidal) limit, are insufficient here, because they assume the dominance of self-gravity in binding a cloud, and neglect external agents, such as pressure and tidal forces, which are common in galactic nuclei. We formulate a new scheme for judging this stability. We first revisit the conventional Virial theorem, taking into account an external pressure, to identify the correct range of masses that lead to stable clouds. We then extend the theorem to include an external tidal field, crucial for the stability in the region of interest -- in dense star clusters, around SMBHs. We apply our extended Virial theorem to find the cor...
Pakmehr, Mehrdad; Fitzgerald, Nathan; Feron, Eric; Shamma, Jeff; Behbahani, Alireza
2012-01-01
This manuscript aims to develop and describe gain scheduling control concept for a gas turbine engine which drives a variable pitch propeller. An architecture for gain-scheduling control is developed that controls the turboshaft engine for large thrust commands in stable fashion with good performance. Fuel ow and propeller pitch angle are the two control inputs of the system. New stability proof has been developed for gain scheduling control of gas turbine engines using global linearization a...
Stability and response bounds of non-conservative linear systems
DEFF Research Database (Denmark)
Kliem, Wolfhard; Pommer, Christian
2004-01-01
This paper develops a stability theorem and response bounds for non-conservative systems of the form MX + (D + G)x + (K + N)x = f(t), with hermitian positive-definite matrices M, D and K, and skew-hermitian matrices G and N. To this end, we first find a Lyapunov function by solving the Lyapunov...... matrix equation. Then, if a system satisfies the condition of the stability theorem, the associated Lyapunov function can be used to obtain response bounds for the norms as well as for the individual coordinates of the solution. Examples from rotor dynamics illustrate the results....
Virial Theorem of Stellar Stability%星体稳定性的维里定理
Institute of Scientific and Technical Information of China (English)
卡里布努尔·库尔班; 高建功
2001-01-01
The importance of the stability of stellar stucture is discussed.The virial theorem of the star in hydrostatic equilibrium state is deduced. This could be useful in qualitative understanding of the star stability.%讨论了星体构型稳定性的重要性,推导了星体处于流体静力学平衡状态下的维里定理,这对定性地理解星体稳定性是很有用的.
Zhao, Xueyan; Deng, Feiqi
2016-07-01
In this paper, a particular property of Lyapunov functions for functional differential equations (FDEs) is developed, that is the direct dependence of the signs of the derivatives of the Lyapunov functions on the initial data. This property implies that the derivatives of the Lyapunov functions for FDEs cannot be guaranteed to be negative definite generally, and then makes the FDEs differ from the ordinary differential equations constitutionally. With this property, we give some enlightenments for the research methods for establishing stability theorems or criteria for FDEs, which may help us to form a common view about the choice of the investigation methods on the stability of FDEs. The conclusion is stated in both the deterministic and stochastic versions. Two illustrative examples are given to show and verify our conclusion through the paper.
Strict Stability Criteria for Impulsive Functional Differential Systems
Directory of Open Access Journals (Sweden)
Kaien Liu
2008-02-01
Full Text Available By using Lyapunov functions and Razumikhin techniques, the strict stability of impulsive functional differential systems is investigated. Some comparison theorems are given by virtue of differential inequalities. The corresponding theorems in the literature can be deduced from our results.
Greenwood, Nigel J C; Gunton, Jenny E
2014-07-01
This study demonstrated the novel application of a "machine-intelligent" mathematical structure, combining differential game theory and Lyapunov-based control theory, to the artificial pancreas to handle dynamic uncertainties. Realistic type 1 diabetes (T1D) models from the literature were combined into a composite system. Using a mixture of "black box" simulations and actual data from diabetic medical histories, realistic sets of diabetic time series were constructed for blood glucose (BG), interstitial fluid glucose, infused insulin, meal estimates, and sometimes plasma insulin assays. The problem of underdetermined parameters was side stepped by applying a variant of a genetic algorithm to partial information, whereby multiple candidate-personalized models were constructed and then rigorously tested using further data. These formed a "dynamic envelope" of trajectories in state space, where each trajectory was generated by a hypothesis on the hidden T1D system dynamics. This dynamic envelope was then culled to a reduced form to cover observed dynamic behavior. A machine-intelligent autonomous algorithm then implemented game theory to construct real-time insulin infusion strategies, based on the flow of these trajectories through state space and their interactions with hypoglycemic or near-hyperglycemic states. This technique was tested on 2 simulated participants over a total of fifty-five 24-hour days, with no hypoglycemic or hyperglycemic events, despite significant uncertainties from using actual diabetic meal histories with 10-minute warnings. In the main case studies, BG was steered within the desired target set for 99.8% of a 16-hour daily assessment period. Tests confirmed algorithm robustness for ±25% carbohydrate error. For over 99% of the overall 55-day simulation period, either formal controller stability was achieved to the desired target or else the trajectory was within the desired target. These results suggest that this is a stable, high
Li, Zhihong; Liu, Lei; Zhu, Quanxin
2016-12-01
This paper studies the mean-square exponential input-to-state stability of delayed Cohen-Grossberg neural networks with Markovian switching. By using the vector Lyapunov function and property of M-matrix, two generalized Halanay inequalities are established. By means of the generalized Halanay inequalities, sufficient conditions are also obtained, which can ensure the exponential input-to-state stability of delayed Cohen-Grossberg neural networks with Markovian switching. Two numerical examples are given to illustrate the efficiency of the derived results. Copyright © 2016 Elsevier Ltd. All rights reserved.
DEFF Research Database (Denmark)
Eriksson, Robert
2014-01-01
as possible, and is based on Lyapunov theory considering the nonlinear behavior. The time optimal controller is of a bang-bang type and uses wide-area measurements as feedback signals. Nonlinear simulations are run in the Nordic32 test system implemented in PowerFactory/DIgSILENT with an interface to Matlab...
Institute of Scientific and Technical Information of China (English)
岑丽辉; 席裕庚
2009-01-01
A Lyapunov function is constructed based on the weighted sum of entropies for the case of two open channels in cascade, which is described by the Saint-Venant equations. A class of boundary feedback controllers is presented to guaran-tee the local closed-loop asymptotic stability in a neighborhood of the equilibrium point by means of the Lyapunov design ap-proach.
Mazinan, A H
2016-03-01
The research addresses a Lyapunov-based constrained control strategy to deal with the autonomous space system in the presence of large disturbances. The aforementioned autonomous space system under control is first represented through a dynamics model and subsequently the proposed control strategy is fully investigated with a focus on the three-axis detumbling and the corresponding pointing mode control approaches. The three-axis detumbling mode control approach is designed to deal with the unwanted angular rates of the system to be zero, while the saturations of the actuators are taken into consideration. Moreover, the three-axis pointing mode control approach is designed in the similar state to deal with the rotational angles of the system to be desirable. The contribution of the research is mathematically made to propose a control law in connection with a new candidate of Lyapunov function to deal with the rotational angles and the related angular rates of the present autonomous space system with respect to state-of-the-art. A series of experiments are carried out to consider the efficiency of the proposed control strategy, as long as a number of benchmarks are realized in the same condition to verify and guarantee the strategy performance in both modes of control approaches.
Theory and application of stability for stochastic reaction diffusion systems
Institute of Scientific and Technical Information of China (English)
LUO Qi; DENG FeiQi; MAO XueRong; BAO JunDong; ZHANG YuTian
2008-01-01
So far, the Lyapunov direct method is still the moat effective technique in the study of stability for ordinary differential equations and stochastic differential equations. Due to the shortage of the corresponding Ito formula, this useful method has not been popularized in stochastic partial differential equations. The aim of this work is to try to extend the Lyapunov direct method to the Ito stochastic reaction diffusion systems and to establish the corresponding Lyapunov stability theory, including stability in probablity, asymptotic stability in probablity, end exponential stability in mean square. As the application of the obtained theorems, this paper addresses the stability of the Hopfield neural network and points out that the main results ob-tained by Holden Helge and Liao Xiaoxin et al. can be all regarded as the corollaries of the theorems presented in this paper.
Coordinate-invariant incremental Lyapunov functions
Zamani, Majid
2011-01-01
The notion of incremental stability was proposed by several researchers as a strong property of dynamical and control systems. In this type of stability, the focus is on the convergence of trajectories with respect to themselves, rather than with respect to an equilibrium point or a particular trajectory. Similarly to stability, Lyapunov functions play an important role in the study of incremental stability. In this paper, we propose coordinate-invariant notions of incremental Lyapunov function and provide the description of incremental stability in terms of existence of the proposed Lyapunov functions. Moreover, we develop a backstepping design approach providing a recursive way of constructing controllers as well as incremental Lyapunov functions. The effectiveness of our method is illustrated by synthesizing a controller rendering a single-machine infinite-bus electrical power system incrementally stable.
Robust lyapunov controller for uncertain systems
Laleg-Kirati, Taous-Meriem
2017-02-23
Various examples of systems and methods are provided for Lyapunov control for uncertain systems. In one example, a system includes a process plant and a robust Lyapunov controller configured to control an input of the process plant. The robust Lyapunov controller includes an inner closed loop Lyapunov controller and an outer closed loop error stabilizer. In another example, a method includes monitoring a system output of a process plant; generating an estimated system control input based upon a defined output reference; generating a system control input using the estimated system control input and a compensation term; and adjusting the process plant based upon the system control input to force the system output to track the defined output reference. An inner closed loop Lyapunov controller can generate the estimated system control input and an outer closed loop error stabilizer can generate the system control input.
Extensions of the stability theorem of the Minkowski space in general relativity
Bieri, Lydia
2009-01-01
A famous result of Christodoulou and Klainerman is the global nonlinear stability of Minkowski spacetime. In this book, Bieri and Zipser provide two extensions to this result. In the first part, Bieri solves the Cauchy problem for the Einstein vacuum equations with more general, asymptotically flat initial data, and describes precisely the asymptotic behavior. In particular, she assumes less decay in the power of r and one less derivative than in the Christodoulou-Klainerman result. She proves that in this case, too, the initial data, being globally close to the trivial data, yields a solution which is a complete spacetime, tending to the Minkowski spacetime at infinity along any geodesic. In contrast to the original situation, certain estimates in this proof are borderline in view of decay, indicating that the conditions in the main theorem on the decay at infinity on the initial data are sharp. In the second part, Zipser proves the existence of smooth, global solutions to the Einstein-Maxwell equations. A n...
Lyapunov inequalities for the periodic boundary value problem at higher eigenvalues
Canada, Antonio
2009-01-01
This paper is devoted to provide some new results on Lyapunov type inequalities for the periodic boundary value problem at higher eigenvalues. Our main result is derived from a detailed analysis on the number and distribution of zeros of nontrivial solutions and their first derivatives, together with the study of some special minimization problems, where the Lagrange multiplier Theorem plays a fundamental role. This allows to obtain the optimal constants. Our applications include the Hill's equation where we give some new conditions on its stability properties and also the study of periodic and nonlinear problems at resonance where we show some new conditions which allow to prove the existence and uniqueness of solutions.
Random Matrices and Lyapunov Coefficients Regularity
Gallavotti, Giovanni
2017-02-01
Analyticity and other properties of the largest or smallest Lyapunov exponent of a product of real matrices with a "cone property" are studied as functions of the matrices entries, as long as they vary without destroying the cone property. The result is applied to stability directions, Lyapunov coefficients and Lyapunov exponents of a class of products of random matrices and to dynamical systems. The results are not new and the method is the main point of this work: it is is based on the classical theory of the Mayer series in Statistical Mechanics of rarefied gases.
Upper quantum Lyapunov exponent and parametric oscillators
Jauslin, H. R.; Sapin, O.; Guérin, S.; Wreszinski, W. F.
2004-11-01
We introduce a definition of upper Lyapunov exponent for quantum systems in the Heisenberg representation, and apply it to parametric quantum oscillators. We provide a simple proof that the upper quantum Lyapunov exponent ranges from zero to a positive value, as the parameters range from the classical system's region of stability to the instability region. It is also proved that in the instability region the parametric quantum oscillator satisfies the discrete quantum Anosov relations defined by Emch, Narnhofer, Sewell, and Thirring.
Exponential Stability of Uncertain T-S Fuzzy Switched Systems with Time Delay
Institute of Scientific and Technical Information of China (English)
Fatima Ahmida; El Houssaine Tissir
2013-01-01
This paper discusses the delay-dependent exponential stability of a class of uncertain T-S fuzzy switched systems with time delay.The method is based on Lyapunov stability theorem and free weighting matrices approach.Two illustrative examples are given to demonstrate the effectiveness of the proposed method.
THE EXISTENCE, UNIQUENESS AND STABILITY OF ALMOST PERIODIC SOLUTION FOR A CLASS OF NONLINEAR SYSTEM
Institute of Scientific and Technical Information of China (English)
方聪娜; 王全义
2004-01-01
In this paper, we study the problems on the existence, uniqueness and stability of almost periodic solution for a class of nonlinear system. Using fixed point theorem and Lyapunov functional, the sufficient conditions are given which guarantee the existence, uniqueness and stability of almost periodic solution for the system.
Kolyada, Sergiy; Rybak, Oleksandr
2013-01-01
We introduce and study the Lyapunov numbers -- quantitative measures of the sensitivity of a dynamical system $(X,f)$ given by a compact metric space $X$ and a continuous map $f:X \\to X$. In particular, we prove that for a minimal topologically weakly mixing system all Lyapunov numbers are the same.
Directory of Open Access Journals (Sweden)
Seung Kwan Song
2016-10-01
Full Text Available We present two control strategies for an oscillating water column-wave energy converter (OWC-WEC in the time domain. We consider a fixed OWC-WEC on the open sea with an impulse turbine module. This system mainly consists of a chamber, turbine and electric generator. For the time domain analysis, all of the conversion stages considering mutualities among them should be analyzed based on the Newtonian mechanics. According to the analysis of Newtonian mechanics, the hydrodynamics of wave energy absorption in the chamber and the turbine aerodynamic performance are directly coupled and share the internal air pressure term via the incompressible air assumption. The turbine aerodynamics and the dynamics of the electric generator are connected by torque load through the rotor shaft, which depends on an electric terminal load that acts as a control input. The proposed control strategies are an instant maximum turbine efficiency tracking control and a constant angular velocity of the turbine rotor control methods. Both are derived by Lyapunov stability analysis. Numerical simulations are carried out under irregular waves with various heights and periods in the time domain, and the results with the controllers are analyzed. We then compare these results with simulations carried out in the absence of the control strategy in order to prove the performance of the controllers.
Beaudette, Shawn M; Howarth, Samuel J; Graham, Ryan B; Brown, Stephen H M
2016-10-01
Several different state-space reconstruction methods have been employed to assess the local dynamic stability (LDS) of a 3D kinematic system. One common method is to use a Euclidean norm (N) transformation of three orthogonal x, y, and z time-series' followed by the calculation of the maximum finite-time Lyapunov exponent (λmax) from the resultant N waveform (using a time-delayed state space reconstruction technique). By essentially acting as a weighted average, N has been suggested to account for simultaneous expansion and contraction along separate degrees of freedom within a 3D system (e.g. the coupling of dynamic movements between orthogonal planes). However, when estimating LDS using N, non-linear transformations inherent within the calculation of N should be accounted for. Results demonstrate that the use of N on 3D time-series data with arbitrary magnitudes of relative bias and zero-crossings cause the introduction of error in estimates of λmax obtained through N. To develop a standard for the analysis of 3D dynamic kinematic waveforms, we suggest that all dimensions of a 3D signal be independently shifted to avoid the incidence of zero-crossings prior to the calculation of N and subsequent estimation of LDS through the use of λmax.
Global stabilization of the new chaotic system based on small-gain theorem
Energy Technology Data Exchange (ETDEWEB)
Liu Debin [Department of Mathematics, Huazhong University of Science and Technology, Wuhan 430074 (China); Yang Xiaosong [Department of Mathematics, Huazhong University of Science and Technology, Wuhan 430074 (China)], E-mail: yangxs@cqupt.edu.cn
2008-05-15
In this paper, we study chaos control of the new chaotic system. Based on small-gain theorem, a class of nonlinear feedback controllers are designed such that the equilibria of the controlled systems are globally asymptotically stable, thus eliminating chaos.
ASYMPTOTIC STABILITIES OF STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS
Institute of Scientific and Technical Information of China (English)
SHEN Yi; JIANG Ming-hui; LIAO Xiao-xin
2006-01-01
Asymptotic characteristic of solution of the stochastic functional differential equation was discussed and sufficient condition was established by multiple Lyapunov functions for locating the limit set of t he solution. Moreover, from them many effective criteria on stochastic asymptotic stability, which enable us to construct the Lyapunov functions much more easily in application, were obtained. The results show that the wellknown classical theorem on stochastic asymptotic stability is a special case of our more general results. In the end, application in stochastic Hopfield neural networks is given to verify our results.
Research on the stability of control systems described by fractional-order transfer functions
Institute of Scientific and Technical Information of China (English)
Zeng Qingshan; Zhu Xinjian; Cao Guangyi
2005-01-01
The stability of control systems described by fractional-order transfer function form is mainly investigated. The stability analysis of integer-order linear systems was extended to the fractional-order control systems. The stability definition of fractional-order linear control systems is presented in terms of the Lyapunov's stability theory. Using the theorems of the Mittag-Leffler function in two parameters directly derives the stability conclusion. The illustrative examples are also given by simulation results.
Comparison between covariant and orthogonal Lyapunov vectors.
Yang, Hong-liu; Radons, Günter
2010-10-01
Two sets of vectors, covariant Lyapunov vectors (CLVs) and orthogonal Lyapunov vectors (OLVs), are currently used to characterize the linear stability of chaotic systems. A comparison is made to show their similarity and difference, especially with respect to the influence on hydrodynamic Lyapunov modes (HLMs). Our numerical simulations show that in both Hamiltonian and dissipative systems HLMs formerly detected via OLVs survive if CLVs are used instead. Moreover, the previous classification of two universality classes works for CLVs as well, i.e., the dispersion relation is linear for Hamiltonian systems and quadratic for dissipative systems, respectively. The significance of HLMs changes in different ways for Hamiltonian and dissipative systems with the replacement of OLVs with CLVs. For general dissipative systems with nonhyperbolic dynamics the long-wavelength structure in Lyapunov vectors corresponding to near-zero Lyapunov exponents is strongly reduced if CLVs are used instead, whereas for highly hyperbolic dissipative systems the significance of HLMs is nearly identical for CLVs and OLVs. In contrast the HLM significance of Hamiltonian systems is always comparable for CLVs and OLVs irrespective of hyperbolicity. We also find that in Hamiltonian systems different symmetry relations between conjugate pairs are observed for CLVs and OLVs. Especially, CLVs in a conjugate pair are statistically indistinguishable in consequence of the microreversibility of Hamiltonian systems. Transformation properties of Lyapunov exponents, CLVs, and hyperbolicity under changes of coordinate are discussed in appendices.
Indian Academy of Sciences (India)
P Dutt; Akhlaq Husain; A S Vasudeva Murthy; C S Upadhyay
2015-08-01
This is the second of a series of papers devoted to the study of ℎ- spectral element methods for three dimensional elliptic problems on non-smooth domains. The present paper addresses the proof of the main stability theorem.We assume that the differential operator is a strongly elliptic operator which satisfies Lax–Milgram conditions. The spectral element functions are non-conforming. The stability estimate theorem of this paper will be used to design a numerical scheme which give exponentially accurate solutions to three dimensional elliptic problems on non-smooth domains and can be easily implemented on parallel computers.
Discussion of stability in a class of models on recurrent wavelet neural networks
Institute of Scientific and Technical Information of China (English)
DENG Ren; LI Zhu-xin; FAN You-hong
2007-01-01
Based on wavelet neural networks (WNNs) and recurrent neural networks (RNNs), a class of models on recurrent wavelet neural networks (RWNNs) is proposed.The new networks possess the advantages of WNNs and RNNs. In this paper, asymptotic stability of RWNNs is researched according to the Lyapunov theorem, and some theorems and formulae are given. The simulation results show the excellent performance of the networks in nonlinear dynamic system recognition.
Lyapunov稳定性逆定理的另一种证明%Another Proof of the Converse Lyapunov Stability Theorem
Institute of Scientific and Technical Information of China (English)
郭晓丽; 方建印
2004-01-01
Lyapunov方法是研究非线性控制问题的重要手段,而Lyapunov稳定性逆定理又是应用此方法的主要定理,但其证明却非常烦琐.利用直接取指数函数的方法,简化了Lyapunov稳定性逆定理中关键函数G的构造,给出了Lyapunov稳定性逆定理的另一种简单的证明方法.
Lyapunov Exponents and Covariant Vectors for Turbulent Flow Simulations
Blonigan, Patrick; Murman, Scott; Fernandez, Pablo; Wang, Qiqi
2016-11-01
As computational power increases, engineers are beginning to use scale-resolving turbulent flow simulations for applications in which jets, wakes, and separation dominate. However, the chaotic dynamics exhibited by scale-resolving simulations poses problems for the conventional sensitivity analysis and stability analysis approaches that are vital for design and control. Lyapunov analysis is used to study the chaotic behavior of dynamical systems, including flow simulations. Lyapunov exponents are the growth or a decay rate of specific flow field perturbations called the Lyapunov covariant vectors. Recently, the authors have used Lyapunov analysis to study the breakdown in conventional sensitivity analysis and the cost of new shadowing-based sensitivity analysis. The current work reviews Lyapunov analysis and presents new results for a DNS of turbulent channel flow, wall-modeled channel flow, and a DNS of a low pressure turbine blade. Additionally, the implications of these Lyapunov analyses for computing sensitivities of these flow simulations will be discussed.
Construction of Lyapunov functions by the localization method
Krishchenko, A. P.; Kanatnikov, A. N.
2017-07-01
In this paper, we examine the problem of construction of Lyapunov functions for asymptotically stable equilibrium points. We exploit conditions of asymptotic stability in terms of compact invariant sets and positively invariant sets. Our results are methods of verification of these conditions and construction of Lyapunov functions by the localization method of compact invariant sets. These results are illustrated by an example.
PERMANENCE AND GLOBAL STABILITY OF A FEEDBACK CONTROL SYSTEM WITH DELAYS
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
This paper considers a feedback control systems of differential equations with delays. By applying the differential inequality theorem, sufficient conditions for the permanence of the system are obtained. Also, by constructing a suitable Lyapunov functional, a criterion for the global stability of the model is obtained.
Institute of Scientific and Technical Information of China (English)
ZHU Qing; LIANG Fang; ZHANG Qing
2009-01-01
In this paper, the Cohen-Grossberg neural networks with time-varying delays and impulses are considered. New sufficient conditions for the existence and global exponential stability of a unique equilibrium point are established by using the fixed point theorem and Lyapunov functional. An example is given to demonstrate the effectiveness of our results.
On Polynomial Stability of Variational Nonautonomous Difference Equations in Banach Spaces
Directory of Open Access Journals (Sweden)
Mihail Megan
2013-01-01
Full Text Available Our goal in this paper is to give characterizations for some concepts of polynomial stability for variational nonautonomous difference equations. The obtained results can be considered generalizations for the case of variational nonautonomous difference equations of some theorems proved by Barbashin (1967, Datko (1973, and Lyapunov (1992, for evolution operators.
Lyapunov functions for fractional order systems
Aguila-Camacho, Norelys; Duarte-Mermoud, Manuel A.; Gallegos, Javier A.
2014-09-01
A new lemma for the Caputo fractional derivatives, when 0<α<1, is proposed in this paper. This result has proved to be useful in order to apply the fractional-order extension of Lyapunov direct method, to demonstrate the stability of many fractional order systems, which can be nonlinear and time varying.
Institute of Scientific and Technical Information of China (English)
Cuimei ZHANG; Wencheng CHEN; Yu YANG
2006-01-01
In this paper, we study the existence and global asymptotic stability of positive periodic solutions of a delayed periodic predator-prey system with Holling Ⅱ type functional response. By use of the continuation theorem of coincidence degree theory and the method of Lyapunov function, some sufficient conditions are obtained.
Stochastic stability properties of jump linear systems
Feng, Xiangbo; Loparo, Kenneth A.; Ji, Yuandong; Chizeck, Howard J.
1992-01-01
Jump linear systems are defined as a family of linear systems with randomly jumping parameters (usually governed by a Markov jump process) and are used to model systems subject to failures or changes in structure. The authors study stochastic stability properties in jump linear systems and the relationship among various moment and sample path stability properties. It is shown that all second moment stability properties are equivalent and are sufficient for almost sure sample path stability, and a testable necessary and sufficient condition for second moment stability is derived. The Lyapunov exponent method for the study of almost sure sample stability is discussed, and a theorem which characterizes the Lyapunov exponents of jump linear systems is presented.
On the Stabilization of the Inverted-Cart Pendulum Using the Saturation Function Approach
Directory of Open Access Journals (Sweden)
Carlos Aguilar-Ibañez
2011-01-01
of a well-characterized small vicinity, whereas the second control action asymptotically brings the whole state of the system to the origin. The corresponding closed-loop stability analysis uses standard linear stability arguments as well as the traditional Lyapunov method and the LaSalle's theorem. Our proposed control law ensures global stability of the system in the upper half plane. We illustrate the effectiveness of the proposed control strategy via numerical simulations.
Directory of Open Access Journals (Sweden)
Chuangxia Huang
2011-01-01
Full Text Available Stability of reaction-diffusion recurrent neural networks (RNNs with continuously distributed delays and stochastic influence are considered. Some new sufficient conditions to guarantee the almost sure exponential stability and mean square exponential stability of an equilibrium solution are obtained, respectively. Lyapunov's functional method, M-matrix properties, some inequality technique, and nonnegative semimartingale convergence theorem are used in our approach. The obtained conclusions improve some published results.
Fuzzy Lyapunov Reinforcement Learning for Non Linear Systems.
Kumar, Abhishek; Sharma, Rajneesh
2017-03-01
We propose a fuzzy reinforcement learning (RL) based controller that generates a stable control action by lyapunov constraining fuzzy linguistic rules. In particular, we attempt at lyapunov constraining the consequent part of fuzzy rules in a fuzzy RL setup. Ours is a first attempt at designing a linguistic RL controller with lyapunov constrained fuzzy consequents to progressively learn a stable optimal policy. The proposed controller does not need system model or desired response and can effectively handle disturbances in continuous state-action space problems. Proposed controller has been employed on the benchmark Inverted Pendulum (IP) and Rotational/Translational Proof-Mass Actuator (RTAC) control problems (with and without disturbances). Simulation results and comparison against a) baseline fuzzy Q learning, b) Lyapunov theory based Actor-Critic, and c) Lyapunov theory based Markov game controller, elucidate stability and viability of the proposed control scheme.
OBSERVING LYAPUNOV EXPONENTS OF INFINITE-DIMENSIONAL DYNAMICAL SYSTEMS.
Ott, William; Rivas, Mauricio A; West, James
2015-12-01
Can Lyapunov exponents of infinite-dimensional dynamical systems be observed by projecting the dynamics into ℝ (N) using a 'typical' nonlinear projection map? We answer this question affirmatively by developing embedding theorems for compact invariant sets associated with C(1) maps on Hilbert spaces. Examples of such discrete-time dynamical systems include time-T maps and Poincaré return maps generated by the solution semigroups of evolution partial differential equations. We make every effort to place hypotheses on the projected dynamics rather than on the underlying infinite-dimensional dynamical system. In so doing, we adopt an empirical approach and formulate checkable conditions under which a Lyapunov exponent computed from experimental data will be a Lyapunov exponent of the infinite-dimensional dynamical system under study (provided the nonlinear projection map producing the data is typical in the sense of prevalence).
Complementarity Properties of the Lyapunov Transformation over Symmetric Cones
Institute of Scientific and Technical Information of China (English)
Yuan Min LI; Xing Tao WANG; De Yun WEI
2012-01-01
The well-known Lyapunov's theorem in matrix theory/continuous dynamical systems asserts that a square matrix A is positive stable if and only if there exists a positive definite matrix X such that AX+XA* is positive definite.In this paper,we extend this theorem to the setting of any Euclidean Jordan algebra V.Given any element a ∈ V,we consider the corresponding Lyapunov transformation La and show that the P and S-properties are both equivalent to a being positive. Then we characterize the Ro-property for La and show that La has the R0-property if and only if a is invertible.Finally,we provide La with some characterizatious of the E0-property and the nondegeneracy property.
Lyapunov Functions and Solutions of the Lyapunov Matrix Equation for Marginally Stable Systems
DEFF Research Database (Denmark)
Kliem, Wolfhard; Pommer, Christian
2000-01-01
of the Lyapunov matrix equation and characterize the set of matrices $(B, C)$ which guarantees marginal stability. The theory is applied to gyroscopic systems, to indefinite damped systems, and to circulatory systems, showing how to choose certain parameter matrices to get sufficient conditions for marginal...
A Lyapunov-Based Extension to Particle Swarm Dynamics for Continuous Function Optimization
Bhattacharya, Sayantani; Konar, Amit; Das, Swagatam; Han, Sang Yong
2009-01-01
The paper proposes three alternative extensions to the classical global-best particle swarm optimization dynamics, and compares their relative performance with the standard particle swarm algorithm. The first extension, which readily follows from the well-known Lyapunov's stability theorem, provides a mathematical basis of the particle dynamics with a guaranteed convergence at an optimum. The inclusion of local and global attractors to this dynamics leads to faster convergence speed and better accuracy than the classical one. The second extension augments the velocity adaptation equation by a negative randomly weighted positional term of individual particle, while the third extension considers the negative positional term in place of the inertial term. Computer simulations further reveal that the last two extensions outperform both the classical and the first extension in terms of convergence speed and accuracy. PMID:22303158
Stability Analysis of Continuous-Time Fuzzy Large-Scale System
Institute of Scientific and Technical Information of China (English)
曾怡达; 张友刚; 肖建
2003-01-01
A continuous-time fuzzy large-scale system F consists of some interconnected Takagi-Sugeno fuzzy subsystems. Two sufficient conditions for the asymptotic stability of this system (namely, theorem 1 and theorem 2) are derived via a multiple Lyapunov function approach. In theorem 1, the information of membership functions of fuzzy rules should be known in order to analyze the stability of F. But in general this information is not easy to be acquired for their time-varying property. So theorem 2 is provided to judge the asymptotic stability of F, based on which there is no need to know the information of membership functions in stability analysis. Finally, a numerical example is given to show the utility of the method proposed in this paper.
On Delay Independent Stabilization Analysis for a Class of Switched Large-Scale Time-Delay Systems
Directory of Open Access Journals (Sweden)
Chi-Jo Wang
2013-01-01
Full Text Available In view of the state-driven switching method, the sufficient stability conditions with delay independence will be derived for the switched large-scale time-delay systems. A new stability criterion of switched large-scale time-delay systems is deduced by Lyapunov stability theorem. The method can be applied to cases when all individual switched systems are unstable. Finally, one example is exploited to illustrate the proposed schemes.
Jiao, Ticao; Xu, Shengyuan; Lu, Junwei; Wei, Yunliang; Zou, Yun
2016-01-01
This paper deals with the decentralised output feedback stabilisation problem for a class of large-scale stochastic time-delay nonlinear systems. A general theorem is firstly given to guarantee the global existence and uniqueness of the solution for stochastic time-delay systems. In addition, a stochastic version of the well-known LaSalle-Yoshizawa theorem with time-varying delay is initially proposed for the controller design and stability analysis. Then, for a class of large-scale stochastic systems with time-varying delays, totally decentralised adaptive delay-dependent controllers are designed by using K-filter and backstepping approach. Via LaSalle-Yoshizawa-type theorem and constructing a general Lyapunov function, it is shown that all signals in the closed-loop system are bounded almost surely and the solution is almost surely asymptotically stable. Finally, a simulation example is given to illustrate the effectiveness of the results of this paper.
Directory of Open Access Journals (Sweden)
Jin Zhu
2009-01-01
Full Text Available Switching controller design for a class of Markovian jump nonlinear systems with unmodeled dynamics is considered in this paper. Based on the differential equation and infinitesimal generator of jump systems, the concept of Jump Input-to-State practical Stability (JISpS in probability and stochastic Lyapunov stability criterion are put forward. By using backsetpping technology and stochastic small-gain theorem, a switching controller is proposed which ensures JISpS in probability for the jump nonlinear system. A simulation example illustrates the validity of this design.
Flatto, Leopold
2009-01-01
Poncelet's theorem is a famous result in algebraic geometry, dating to the early part of the nineteenth century. It concerns closed polygons inscribed in one conic and circumscribed about another. The theorem is of great depth in that it relates to a large and diverse body of mathematics. There are several proofs of the theorem, none of which is elementary. A particularly attractive feature of the theorem, which is easily understood but difficult to prove, is that it serves as a prism through which one can learn and appreciate a lot of beautiful mathematics. This book stresses the modern appro
Automatically Discovering Relaxed Lyapunov Functions for Polynomial Dynamical Systems
Liu, Jiang; Zhao, Hengjun
2011-01-01
The notion of Lyapunov function plays a key role in design and verification of dynamical systems, as well as hybrid and cyber-physical systems. In this paper, to analyze the asymptotic stability of a dynamical system, we generalize standard Lyapunov functions to relaxed Lyapunov functions (RLFs), by considering higher order Lie derivatives of certain functions along the system's vector field. Furthermore, we present a complete method to automatically discovering polynomial RLFs for polynomial dynamical systems (PDSs). Our method is complete in the sense that it is able to discover all polynomial RLFs by enumerating all polynomial templates for any PDS.
A normal form theorem around symplectic leaves
Crainic, M.N.; Marcut, I.T.
2012-01-01
We prove the Poisson geometric version of the Local Reeb Stability (from foliation theory) and of the Slice Theorem (from equivariant geometry), which is also a generalization of Conn’s linearization theorem.
Institute of Scientific and Technical Information of China (English)
吴东平
2012-01-01
基于Lyapunov运动稳定性理论,经过推导可知,一个单自由度的某一个受迫振动的特解的运动稳定性问题等价于这个单自由度系统自由振动的稳定问题.对于复杂非线性系统的动力稳定性问题,直接应用Lyapunov理论进行系统的动力稳定性判定比较困难,考虑大跨度拱型结构的变形特征,提出一种简洁、实用且适合数值计算的动力稳定性判别方法——位移时程变化法.运用该方法计算结构在承受一般动荷载类型和不同计算条件下的动力稳定性,验证此方法的实用性及正确性.%Based on the stability theory of Lyapunov motion,it can be derived that the motion stability problem of a particular solution to a forced vibration of a single degree of freedom system,is equal to the stability problem of the free vibration system of single degree of freedom.For a dynamic stability problem of complex nonlinear system,it is difficult to judge the dynamic stability of a system by the theory of Lyapunov motion directly.Considering the deformation feature of a large-span arch structure,a simple and practical theory named deformation history theory which is suitable for digital computation to judge the dynamic stability of an arch structure is put forward and used to judge the dynamic stability of an arch structure under common dynamic loading and different conditions.It is shown that the theory is applicable and reasonable.
Lyapunov Computational Method for Two-Dimensional Boussinesq Equation
Mabrouk, Anouar Ben
2010-01-01
A numerical method is developed leading to Lyapunov operators to approximate the solution of two-dimensional Boussinesq equation. It consists of an order reduction method and a finite difference discretization. It is proved to be uniquely solvable and analyzed for local truncation error for consistency. The stability is checked by using Lyapunov criterion and the convergence is studied. Some numerical implementations are provided at the end of the paper to validate the theoretical results.
A Spectral Lyapunov Function for Exponentially Stable LTV Systems
Zhu, J. Jim; Liu, Yong; Hang, Rui
2010-01-01
This paper presents the formulation of a Lyapunov function for an exponentially stable linear timevarying (LTV) system using a well-defined PD-spectrum and the associated PD-eigenvectors. It provides a bridge between the first and second methods of Lyapunov for stability assessment, and will find significant applications in the analysis and control law design for LTV systems and linearizable nonlinear time-varying systems.
Lyapunov functions for a class of nonlinear systems using Caputo derivative
Fernandez-Anaya, G.; Nava-Antonio, G.; Jamous-Galante, J.; Muñoz-Vega, R.; Hernández-Martínez, E. G.
2017-02-01
This paper presents an extension of recent results that allow proving the stability of Caputo nonlinear and time-varying systems, by means of the fractional order Lyapunov direct method, using quadratic Lyapunov functions. This article introduces a new way of building polynomial Lyapunov functions of any positive integer order as a way of determining the stability of a greater variety of systems when the order of the derivative is 0 < α < 1. Some examples are given to validate these results.
Jumping property of Lyapunov values
Institute of Scientific and Technical Information of China (English)
毛锐; 王铎
1996-01-01
A sufficient condition for fcth Lyapunov value to be zero for planar polynomial vector fields is given, which extends the result of "jumping property’ of Lyapunov values obtained by Wang Duo to more general cases. A concrete example that the origin cannot be weak focus of order 1, 2, 4, 5, 8 is presented.
Institute of Scientific and Technical Information of China (English)
曾庆山; 曹广益; 朱新坚
2004-01-01
The definitions of controllability, observability and stability were presented for fractional-order linear systems. Using the Cayley-Hamilton theorem and Mittag-Leffler function in two parameters, the sufficient and necessary conditions of controllability and observability for such systems were derived. In terms of Lyapunov's stability theory, using the theorems of Mittage-Leffler function in two parameters this paper directly derived the sufficient and necessary condition of stability for such systems. The results obtained are useful for the analysis and synthesis of fractional-order linear control systems.
Robustness and robust stability of the active sliding mode synchronization
Energy Technology Data Exchange (ETDEWEB)
Naseh, Majid Reza [Electrical Engineering Department, Islamic Azad University, Birjand Branch (Iran, Islamic Republic of)], E-mail: naseh@ee.src.aiu.ir; Haeri, Mohammad [Advanced Control System Lab., Electrical Engineering Department, Sharif University Technology, Tehran (Iran, Islamic Republic of)], E-mail: haeri@sina.sharif.edu
2009-01-15
We have developed relations between uncertainties and signals bounds in one side and the control parameters on the other side in the case of the active sliding mode synchronization. Using Lyapunov stability theorem, we have determined uncertainties levels for which synchronization is achieved for a given set of the control parameters. We have run a nonlinear programming algorithm to determine the control parameters for specific range of the uncertainties. Finally, numerical simulations are presented to verify the derived relations.
Heck, Richard G
2011-01-01
Frege's Theorem collects eleven essays by Richard G Heck, Jr, one of the world's leading authorities on Frege's philosophy. The Theorem is the central contribution of Gottlob Frege's formal work on arithmetic. It tells us that the axioms of arithmetic can be derived, purely logically, from a single principle: the number of these things is the same as the number of those things just in case these can be matched up one-to-one with those. But that principle seems so utterlyfundamental to thought about number that it might almost count as a definition of number. If so, Frege's Theorem shows that a
Universal construction of control Lyapunov functions for a class of nonlinear systems
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
A method is developed by which control Lyapunov functions of a class of nonlinear systems can be constructed systematically.Based on the control Lyapunov function,a feedback control is obtained to stabilize the closed-loop system.In addition,this method is applied to stabilize the Benchmark system.A simulation shows the effectiveness of the method.
Lyapunov modes in extended systems.
Yang, Hong-Liu; Radons, Günter
2009-08-28
Hydrodynamic Lyapunov modes, which have recently been observed in many extended systems with translational symmetry, such as hard sphere systems, dynamic XY models or Lennard-Jones fluids, are nowadays regarded as fundamental objects connecting nonlinear dynamics and statistical physics. We review here our recent results on Lyapunov modes in extended system. The solution to one of the puzzles, the appearance of good and 'vague' modes, is presented for the model system of coupled map lattices. The structural properties of these modes are related to the phase space geometry, especially the angles between Oseledec subspaces, and to fluctuations of local Lyapunov exponents. In this context, we report also on the possible appearance of branches splitting in the Lyapunov spectra of diatomic systems, similar to acoustic and optical branches for phonons. The final part is devoted to the hyperbolicity of partial differential equations and the effective degrees of freedom of such infinite-dimensional systems.
Lyapunov decay in quantum irreversibility.
García-Mata, Ignacio; Roncaglia, Augusto J; Wisniacki, Diego A
2016-06-13
The Loschmidt echo--also known as fidelity--is a very useful tool to study irreversibility in quantum mechanics due to perturbations or imperfections. Many different regimes, as a function of time and strength of the perturbation, have been identified. For chaotic systems, there is a range of perturbation strengths where the decay of the Loschmidt echo is perturbation independent, and given by the classical Lyapunov exponent. But observation of the Lyapunov decay depends strongly on the type of initial state upon which an average is carried out. This dependence can be removed by averaging the fidelity over the Haar measure, and the Lyapunov regime is recovered, as has been shown for quantum maps. In this work, we introduce an analogous quantity for systems with infinite dimensional Hilbert space, in particular the quantum stadium billiard, and we show clearly the universality of the Lyapunov regime.
Nonlinear Direct Robust Adaptive Control Using Lyapunov Method
Directory of Open Access Journals (Sweden)
Chunbo Xiu
2013-07-01
Full Text Available The problem of robust adaptive stabilization of a class of multi-input nonlinear systems with arbitrary unknown parameters and unknown structure of bounded variation have been considered. By employing the direct adaptive and control Lyapunov function method, a robust adaptive controller is designed to complete the globally adaptive stability of the system states. By employing our result, a kind of nonlinear system is analyzed, the concrete form of the control law is given and the meaningful quadratic control Lyapunov function for the system is constructed. Simulation of parallel manipulator is provided to illustrate the effectiveness of the proposed method.
Oseledec multiplicative ergodic theorem for laminations
Nguyên, Viêt-Anh
2017-01-01
Given a n-dimensional lamination endowed with a Riemannian metric, the author introduces the notion of a multiplicative cocycle of rank d, where n and d are arbitrary positive integers. The holonomy cocycle of a foliation and its exterior powers as well as its tensor powers provide examples of multiplicative cocycles. Next, the author defines the Lyapunov exponents of such a cocycle with respect to a harmonic probability measure directed by the lamination. He also proves an Oseledec multiplicative ergodic theorem in this context. This theorem implies the existence of an Oseledec decomposition almost everywhere which is holonomy invariant. Moreover, in the case of differentiable cocycles the author establishes effective integral estimates for the Lyapunov exponents. These results find applications in the geometric and dynamical theory of laminations. They are also applicable to (not necessarily closed) laminations with singularities. Interesting holonomy properties of a generic leaf of a foliation are obtained...
Directory of Open Access Journals (Sweden)
Yang Chen
2017-03-01
Full Text Available This article proposes a composite path following controller that allows the small fixed-wing unmanned aerial vehicle to follow a predefined path. Assuming that the vehicle is equipped with an autopilot for altitude and airspeed maintained well, the controller design adopts the hierarchical control structure. With the inner-loop controller design based on the notion of active disturbance rejection control which will respond to the desired roll angle command, the core part of the outer-loop controller is designed based on Lyapunov stability theorem to generate the desired course rate for the straight-line paths. The bank to turn maneuver is used to transform the desired course rate to the desired roll angle command. Both the hardware-in-the-loop simulation in the X-Plane simulator and actual experimental flight tests have been successfully achieved, which verified the effectiveness of the proposed method.
Analisis Kestabilan Model Matematika Penyakit Leukimia dengan Fungsi Lyapunov
2015-01-01
This study aims to analyze the stability of the equilibrium point of the mathematical model of leukemia before and after undergoing chemotherapy. Analysis of the stability of the model is done by analyzing the model by using a Lyapunov function. By using MATLAB program will be described stability of the model before chemotherapy and after chemotherapy. The results showed that the equilibrium point of stem cell compartment model is asymptotically stable for certain parameter values. This is be...
Generalized stability of motion of impulsive Lurie-Postnikov systems with structural perturbation
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A. A. Martynyuk
1998-01-01
Full Text Available This paper investigates the absolute stability on ℊs of the zero solution of Lurie-Postnikov systems with impulses and structural perturbation. A number of absolutely stable on ℊs theorems of the Lyapunov type for Lurie-Postnikov systems are proved, extending and generalizing previous work on the subject. These results are applied to some fourth-order Lurie-Postnikov type systems decomposed into two systems.
Stabilization and trajectory tracking of autonomous airship's planar motion
Institute of Scientific and Technical Information of China (English)
Zhang Yan; Qu Weidong; Xi Yugeng; Cai Zili
2008-01-01
The stabilization and trajectory tracking problems of autonomous airship's planar motion are studied.By denning novel configuration error and velocity error,the dynamics of error systems are derived.By applying Lyapunov stability method,the state feedback control laws are designed and the close-loop error systems are proved to be uniformly asymptotically stable by Matrosov theorem.In particular,the controller does not need knowledge on system parameters in the case of set-point stabilization,which makes the controller robust with respect to parameter uncertainty.Numerical simulations illustrate the effectiveness of the controller designed.
Entanglement production and Lyapunov exponents
Hackl, Lucas; Bianchi, Eugenio; Yokomizo, Nelson
2017-01-01
Squeezed vacua play a prominent role in quantum field theory in curved spacetime. Instabilities and resonances that arise from the coupling in the field to the background geometry, result in a large squeezing of the vacuum. In this talk, I discuss the relation between squeezing and Lyapunov exponents of the system. In particular, I derive a new formula for the rate of growth of the entanglement entropy expressed as the sum of the Lyapunov exponents. Examples of such a linear production regime can be found during inflation and in the preheating phase directly after inflation.
La Salle定理的推广%GENERALIZATION OF LA SALLE'S THEOREM
Institute of Scientific and Technical Information of China (English)
甘作新; 韩京清; 王培光
2001-01-01
In this paper, by using the method of Lyapunov function we generalize the theorem of La Salle from Lurie control system to general Lurie control system. Our result is concise and is easy to be tested. Hence, we improve the proof and extend the applied range of La Salle's Theorem.
On the Computation of Lyapunov Functions for Interconnected Systems
DEFF Research Database (Denmark)
Sloth, Christoffer
2016-01-01
This paper addresses the computation of additively separable Lyapunov functions for interconnected systems. The presented results can be applied to reduce the complexity of the computations associated with stability analysis of large scale systems. We provide a necessary and sufficient condition...
Parameter-dependent Lyapunov functional for systems with multiple time delays
Institute of Scientific and Technical Information of China (English)
Min WU; Yong HE
2004-01-01
The separation of the Lyapunov matrices and system matrices plays an important role when one uses parameter-dependent Lyapunov functional handling systems with polytopic type uncertainties.The delay-dependent robust stability problem for systems with polytopic type uncertainties is discussed by using parameter-dependent Lyapunov functional.The derivative term in the derivative of Lyapunov functional is reserved and the free weighting matrices are employed to express the relationship between the terms in the system equation such that the Lyapunov matrices are not involved in any product terms with the system matrices.In addition,the relationships between the terms in the Leibniz Newton formula are also described by some free weighting matrices and some delay-dependent stability conditions are derived.Numerical examples demonstrate that the proposed criteria are more effective than the previous results.
Exponential Stability of Stochastic Nonlinear Dynamical Price System with Delay
Directory of Open Access Journals (Sweden)
Wenli Zhu
2013-01-01
Full Text Available Based on Lyapunov stability theory, Itô formula, stochastic analysis, and matrix theory, we study the exponential stability of the stochastic nonlinear dynamical price system. Using Taylor's theorem, the stochastic nonlinear system with delay is reduced to an n-dimensional semilinear stochastic differential equation with delay. Some sufficient conditions of exponential stability and corollaries for such price system are established by virtue of Lyapunov function. The time delay upper limit is solved by using our theoretical results when the system is exponentially stable. Our theoretical results show that if the classical price Rayleigh equation is exponentially stable, so is its perturbed system with delay provided that both the time delay and the intensity of perturbations are small enough. Two examples are presented to illustrate our results.
Computing Lyapunov spectra with continuous Gram-Schmidt orthonormalization
Christiansen, F; Christiansen, Freddy; Rugh, Hans Henrik
1996-01-01
We present a straightforward and reliable continuous method for computing the full or a partial Lyapunov spectrum associated with a dynamical system specified by a set of differential equations. We do this by introducing a stability parameter beta>0 and augmenting the dynamical system with an orthonormal k-dimensional frame and a Lyapunov vector such that the frame is continuously Gram-Schmidt orthonormalized and at most linear growth of the dynamical variables is involved. We prove that the method is strongly stable when beta > -lambda_k where lambda_k is the k'th Lyapunov exponent in descending order and we show through examples how the method is implemented. It extends many previous results.
Institute of Scientific and Technical Information of China (English)
无
2012-01-01
In this paper,a class of bidirectional associative memory(BAM) recurrent neural networks with delays are studied.By a fixed point theorem and a Lyapunov functional,some new sufficient conditions for the existence,uniqueness and global exponential stability of the almost periodic solutions are established.These conditions are easy to be verified and our results complement the previous known results.
Yi-You Hou; Zhang-Lin Wan
2014-01-01
This paper considers the problem of the robust stability for the nonlinear system with time-varying delay and parameters uncertainties. Based on the H∞ theorem, Lyapunov-Krasovskii theory, and linear matrix inequality (LMI) optimization technique, the H∞ quasi-sliding mode controller and switching function are developed such that the nonlinear system is asymptotically stable in the quasi-sliding mode and satisfies the disturbance attenuation (H∞-norm performance). The effectiveness and accura...
The relativistic virial theorem and scale invariance
Gaite, Jose
2013-01-01
The virial theorem is related to the dilatation properties of bound states. This is realized, in particular, by the Landau-Lifshitz formulation of the relativistic virial theorem, in terms of the trace of the energy-momentum tensor. We construct a Hamiltonian formulation of dilatations in which the relativistic virial theorem naturally arises as the condition of stability against dilatations. A bound state becomes scale invariant in the ultrarelativistic limit, in which its energy vanishes. However, for very relativistic bound states, scale invariance is broken by quantum effects and the virial theorem must include the energy-momentum tensor trace anomaly. This quantum field theory virial theorem is directly related to the Callan-Symanzik equations. The virial theorem is applied to QED and then to QCD, focusing on the bag model of hadrons. In massless QCD, according to the virial theorem, 3/4 of a hadron mass corresponds to quarks and gluons and 1/4 to the trace anomaly.
An improved robust stability result for uncertain neural networks with multiple time delays.
Arik, Sabri
2014-06-01
This paper proposes a new alternative sufficient condition for the existence, uniqueness and global asymptotic stability of the equilibrium point for the class of delayed neural networks under the parameter uncertainties of the neural system. The existence and uniqueness of the equilibrium point is proved by using the Homomorphic mapping theorem. The asymptotic stability of the equilibrium point is established by employing the Lyapunov stability theorems. The obtained robust stability condition establishes a new relationship between the network parameters of the system. We compare our stability result with the previous corresponding robust stability results derived in the past literature. Some comparative numerical examples together with some simulation results are also given to show the applicability and advantages of our result.
Baire classes of Lyapunov invariants
Bykov, V. V.
2017-05-01
It is shown that no relations exist (apart from inherent ones) between Baire classes of Lyapunov transformation invariants in the compact- open and uniform topologies on the space of linear differential systems. It is established that if a functional on the space of linear differential systems with the compact-open topology is the repeated limit of a multisequence of continuous functionals, then these can be chosen to be determined by the values of system coefficients on a finite interval of the half-line (one for each functional). It is proved that the Lyapunov exponents cannot be represented as the limit of a sequence of (not necessarily continuous) functionals such that each of these depends only on the restriction of the system to a finite interval of the half-line. Bibliography: 28 titles.
Lyapunov Stability of Complementarity and Extended Systems
Camlibel, M. Kanat; Pang, Jong-Shi; Shen, Jinglai
2006-01-01
A linear complementarity system (LCS) is a piecewise linear dynamical system consisting of a linear time-invariant ordinary differential equation (ODE) parameterized by an algebraic variable that is required to be a solution to a finite-dimensional linear complementarity problem (LCP), whose
Construction of Lyapunov Function for Dissipative Gyroscopic System
Institute of Scientific and Technical Information of China (English)
XU Wei; YUAN Bo; AO Ping
2011-01-01
@@ We introduce a force decomposition to construct a potential function in deterministic dynamics described by ordinary differential equations in the context of dissipative gyroscopic systems.Such a potential function serves as the corresponding Lyapunov function for the dynamics,hence it gives both quantitative and qualitative descriptions for stability of motion.As an example we apply our force decomposition to a four-dimensional dissipative gyroscopic system.We explicitly obtain the potential function for all parameter regimes in the linear limit,including those regimes where the Lyapunov function was previously believed not to exist.%We introduce a force decomposition to construct a potential function in deterministic dynamics described by ordinary differential equations in the context of dissipative gyroscopic systems. Such a potential function serves as the corresponding Lyapunov function for the dynamics, hence it gives both quantitative and qualitative descriptions for stability of motion. As an example we apply our force decomposition to a four-dimensional dissipative gyroscopic system. We explicitly obtain the potential function for all parameter regimes in the linear limit, including those regimes where the Lyapunov function was previously believed not to exist.
Linearized asymptotic stability for fractional differential equations
Directory of Open Access Journals (Sweden)
Nguyen Cong
2016-06-01
Full Text Available We prove the theorem of linearized asymptotic stability for fractional differential equations. More precisely, we show that an equilibrium of a nonlinear Caputo fractional differential equation is asymptotically stable if its linearization at the equilibrium is asymptotically stable. As a consequence we extend Lyapunov's first method to fractional differential equations by proving that if the spectrum of the linearization is contained in the sector $\\{\\lambda \\in \\mathbb{C} : |\\arg \\lambda| > \\frac{\\alpha \\pi}{2}\\}$ where $\\alpha > 0$ denotes the order of the fractional differential equation, then the equilibrium of the nonlinear fractional differential equation is asymptotically stable.
The quantitative Morse theorem
Loi, Ta Le; Phien, Phan
2013-01-01
In this paper, we give a proof of the quantitative Morse theorem stated by {Y. Yomdin} in \\cite{Y1}. The proof is based on the quantitative Sard theorem, the quantitative inverse function theorem and the quantitative Morse lemma.
Robust Backstepping Control Based on a Lyapunov Redesign for Skid-Steered Wheeled Mobile Robots
Directory of Open Access Journals (Sweden)
Eun-Ju Hwang
2013-01-01
Full Text Available This paper represents a robust backstepping tracking control based on a Lyapunov redesign for Skid‐Steered Wheeled Mobile Robots (WMRs. We present kinematic and dynamic models that explicitly relate the perturbations to the skidding in order to improve the tracking performance during real running. A robust controller is synthesized in the backstepping approach and the Lyapunov redesign technique, which forces the error dynamics to stabilize to the reference trajectories. We design an additional feedback control ‐ a Lyapunov redesign ‐ such that the overall control stabilizes the actual system in the presence of uncertainty and perturbation with the knowledge of the Lyapunov function. Simulation results are provided to validate and analyse the performance and stability of the proposed controller.
Robust stability analysis of uncertain discrete-time systems with state delay
Institute of Scientific and Technical Information of China (English)
任正云; 张立群; 邵惠鹤
2004-01-01
The sufficient conditions of stability for uncertain discrete-time systems with state delay have been proposed by some researchers in the past few years, yet these results may be conservative in application. The stability analysis of these systems is discussed, and the necessary and sufficient condition of stability is derived by method other than constructing Lyapunov function and solving Riccati inequality. The root locations of system characteristic polynomial, which is obtained by augmentation approach and Laplace expansion, determine the stability of uncertain discrete-time systems with state delay, the system is stable if and only if all roots lie within the unit circle. In order to analyze robust stability of system characteristic polynomial effectively, Kharitonov theorem and edge theorem are applied. Example shows the practicability of these methods.
STABILITY OF N-DIMENSIONAL LINEAR SYSTEMS WITH MULTIPLE DELAYS AND APPLICATION TO SYNCHRONIZATION
Institute of Scientific and Technical Information of China (English)
Weihua DENG; Jinhu L(U); Changpin LI
2006-01-01
This paper further investigates the stability of the n-dimensional linear systems with multiple delays. Using Laplace transform, we introduce a definition of characteristic equation for the n-dimensional linear systems with multiple delays. Moreover, one sufficient condition is attained for the Lyapunov globally asymptotical stability of the general multi-delay linear systems. In particular, our result shows that some uncommensurate linear delays systems have the similar stability criterion as that of the commensurate linear delays systems. This result also generalizes that of Chen and Moore (2002). Finally, this theorem is applied to chaos synchronization of the multi-delay coupled Chua's systems.
Indian Academy of Sciences (India)
P Dutt; Akhlaq Husain; A S Vasudeva Murthy; C S Upadhyay
2015-05-01
This is the first of a series of papers devoted to the study of ℎ- spectral element methods for solving three dimensional elliptic boundary value problems on non-smooth domains using parallel computers. In three dimensions there are three different types of singularities namely; the vertex, the edge and the vertex-edge singularities. In addition, the solution is anisotropic in the neighbourhoods of the edges and vertex-edges. To overcome the singularities which arise in the neighbourhoods of vertices, vertex-edges and edges, we use local systems of coordinates. These local coordinates are modified versions of spherical and cylindrical coordinate systems in their respective neighbourhoods. Away from these neighbourhoods standard Cartesian coordinates are used. In each of these neighbourhoods we use a geometrical mesh which becomes finer near the corners and edges. The geometrical mesh becomes a quasi-uniform mesh in the new system of coordinates. We then derive differentiability estimates in these new set of variables and state our main stability estimate theorem using a non-conforming ℎ- spectral element method whose proof is given in a separate paper.
Using Lyapunov function to design optimal controller for AQM routers
Institute of Scientific and Technical Information of China (English)
ZHANG Peng; YE Cheng-qing; MA Xue-ying; CHEN Yan-hua; LI Xin
2007-01-01
It was shown that active queue management schemes implemented in the routers of communication networks supporting transmission control protocol (TCP) flows can be modelled as a feedback control system. In this paper based on Lyapunov function we developed an optimal controller to improve active queue management (AQM) router's stability and response time,which are often in conflict with each other in system performance. Ns-2 simulations showed that optimal controller outperforms PI controller significantly.
Bohmian quantum mechanical and classical Lyapunov exponents for kicked rotor
Energy Technology Data Exchange (ETDEWEB)
Zheng Yindong [Department of Physics, University of North Texas, Denton, TX 76203-1427 (United States); Kobe, Donald H. [Department of Physics, University of North Texas, Denton, TX 76203-1427 (United States)], E-mail: kobe@unt.edu
2008-04-15
Using de Broglie-Bohm approach to quantum theory, we show that the kicked rotor at quantum resonance exhibits quantum chaos for the control parameter K above a threshold. Lyapunov exponents are calculated from the method of Benettin et al. for bounded systems for both the quantum and classical kicked rotor. In the chaotic regime we find stability regions for control parameters equal to even and odd multiples of {pi}, but the quantum regions are only remnants of the classical ones.
MIMO Lyapunov Theory-Based RBF Neural Classifier for Traffic Sign Recognition
Directory of Open Access Journals (Sweden)
King Hann Lim
2012-01-01
Full Text Available Lyapunov theory-based radial basis function neural network (RBFNN is developed for traffic sign recognition in this paper to perform multiple inputs multiple outputs (MIMO classification. Multidimensional input is inserted into RBF nodes and these nodes are linked with multiple weights. An iterative weight adaptation scheme is hence designed with regards to the Lyapunov stability theory to obtain a set of optimum weights. In the design, the Lyapunov function has to be well selected to construct an energy space with a single global minimum. Weight gain is formed later to obey the Lyapunov stability theory. Detail analysis and discussion on the proposed classifier’s properties are included in the paper. The performance comparisons between the proposed classifier and some existing conventional techniques are evaluated using traffic sign patterns. Simulation results reveal that our proposed system achieved better performance with lower number of training iterations.
Institute of Scientific and Technical Information of China (English)
李安; 宋新宇; 王志祥
2011-01-01
该文研究了非线性微分方程关于初始时刻偏差的实用稳定性,利用扰动Lyapunov函数得到了几个非线性动力系统关于初始时刻偏差的实用稳定性准则,所得结论丰富了非线性微分方程关于初始时刻偏差的实用稳定性理论.%In this paper, the practical stability of nonlinear differential equations with solutions starting off with different initial times is investigated. Several practical stability criteria of nonlinear dynamical systems relative to initial time difference are presented by perturbing Lyapunov functions. The results enrich the theory on practical stability of nonlinear differential equations relative to initial time difference.
Lyapunov Function Synthesis - Algorithm and Software
DEFF Research Database (Denmark)
Leth, Tobias; Wisniewski, Rafal; Sloth, Christoffer
2016-01-01
In this paper we introduce an algorithm for the synthesis of polynomial Lyapunov functions for polynomial vector fields. The Lyapunov function is a continuous piecewisepolynomial defined on simplices, which compose a collection of simplices. The algorithm is elaborated and crucial features...
Rank-one LMIs and Lyapunov's inequality
Henrion, D.; Meinsma, Gjerrit
2001-01-01
We describe a new proof of the well-known Lyapunov's matrix inequality about the location of the eigenvalues of a matrix in some region of the complex plane. The proof makes use of standard facts from quadratic and semi-definite programming. Links are established between the Lyapunov matrix,
Rank-one LMIs and Lyapunov's inequality
Henrion, D.; Meinsma, G.
2001-01-01
We describe a new proof of the well-known Lyapunov's matrix inequality about the location of the eigenvalues of a matrix in some region of the complex plane. The proof makes use of standard facts from quadratic and semi-definite programming. Links are established between the Lyapunov matrix, rank-on
Lyapunov Function Synthesis - Algorithm and Software
DEFF Research Database (Denmark)
Leth, Tobias; Sloth, Christoffer; Wisniewski, Rafal
2016-01-01
In this paper we introduce an algorithm for the synthesis of polynomial Lyapunov functions for polynomial vector fields. The Lyapunov function is a continuous piecewisepolynomial defined on simplices, which compose a collection of simplices. The algorithm is elaborated and crucial features...
Yang, Hong-liu; Radons, Günter; Kantz, Holger
2012-12-14
The estimation of Lyapunov exponents from time series suffers from the appearance of spurious Lyapunov exponents due to the necessary embedding procedure. Separating true from spurious exponents poses a fundamental problem which is not yet solved satisfactorily. We show, in this Letter, analytically and numerically that covariant Lyapunov vectors associated with true exponents lie in the tangent space of the reconstructed attractor. Therefore, we use the angle between the covariant Lyapunov vectors and the tangent space of the reconstructed attractor to identify the true Lyapunov exponents. The usefulness of our method, also for noisy situations, is demonstrated by applications to data from model systems and a NMR laser experiment.
Riemannian theory of Hamiltonian chaos and Lyapunov exponents
Casetti, L; Pettini, M; Casetti, Lapo; Clementi, Cecilia; Pettini, Marco
1996-01-01
This paper deals with the problem of analytically computing the largest Lyapunov exponent for many degrees of freedom Hamiltonian systems. This aim is succesfully reached within a theoretical framework that makes use of a geometrization of newtonian dynamics in the language of Riemannian geometry. A new point of view about the origin of chaos in these systems is obtained independently of homoclinic intersections. Chaos is here related to curvature fluctuations of the manifolds whose geodesics are natural motions and is described by means of Jacobi equation for geodesic spread. Under general conditions ane effective stability equation is derived; an analytic formula for the growth-rate of its solutions is worked out and applied to the Fermi-Pasta-Ulam beta model and to a chain of coupled rotators. An excellent agreement is found the theoretical prediction and the values of the Lyapunov exponent obtained by numerical simulations for both models.
Vorticity, Stokes' Theorem and the Gauss's Theorem
Narayanan, M.
2004-12-01
Vorticity is a property of the flow of any fluid and moving fluids acquire properties that allow an engineer to describe that particular flow in greater detail. It is important to recognize that mere motion alone does not guarantee that the air or any fluid has vorticity. Vorticity is one of four important quantities that define the kinematic properties of any fluid flow. The Navier-Stokes equations are the foundation of fluid mechanics, and Stokes' theorem is used in nearly every branch of mechanics as well as electromagnetics. Stokes' Theorem also plays a vital role in many secondary theorems such as those pertaining to vorticity and circulation. However, the divergence theorem is a mathematical statement of the physical fact that, in the absence of the creation or destruction of matter, the density within a region of space can change only by having it flow into, or away from the region through its boundary. This is also known as Gauss's Theorem. It should also be noted that there are many useful extensions of Gauss's Theorem, including the extension to include surfaces of discontinuity in V. Mathematically expressed, Stokes' theorem can be expressed by considering a surface S having a bounding curve C. Here, V is any sufficiently smooth vector field defined on the surface and its bounding curve C. Integral (Surface) [(DEL X V)] . dS = Integral (Contour) [V . dx] In this paper, the author outlines and stresses the importance of studying and teaching these mathematical techniques while developing a course in Hydrology and Fluid Mechanics. References Arfken, G. "Gauss's Theorem." 1.11 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 57-61, 1985. Morse, P. M. and Feshbach, H. "Gauss's Theorem." In Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 37-38, 1953. Eric W. Weisstein. "Divergence Theorem." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/DivergenceTheorem.html
Stability and synchronization of memristor-based fractional-order delayed neural networks.
Chen, Liping; Wu, Ranchao; Cao, Jinde; Liu, Jia-Bao
2015-11-01
Global asymptotic stability and synchronization of a class of fractional-order memristor-based delayed neural networks are investigated. For such problems in integer-order systems, Lyapunov-Krasovskii functional is usually constructed, whereas similar method has not been well developed for fractional-order nonlinear delayed systems. By employing a comparison theorem for a class of fractional-order linear systems with time delay, sufficient condition for global asymptotic stability of fractional memristor-based delayed neural networks is derived. Then, based on linear error feedback control, the synchronization criterion for such neural networks is also presented. Numerical simulations are given to demonstrate the effectiveness of the theoretical results.
Global Stability Analysis for an Internet Congestion Control Model with a Time-Varying Link Capacity
Rezaie, B; Analoui, M; Khorsandi, S
2009-01-01
In this paper, a global stability analysis is given for a rate-based congestion control system modeled by a nonlinear delayed differential equation. The model determines the dynamics of a single-source single-link network, with a time-varying capacity of link and a fixed communication delay. We obtain a sufficient delay-independent conditions on system parameters under which global asymptotic stability of the system is guarantied. The proof is based on an extension of Lyapunov-Krasovskii theorem for a class of nonlinear time-delay systems. The numerical simulations for a typical scenario justify the theoretical results.
Institute of Scientific and Technical Information of China (English)
阮炯; 王军平; 郭德典
2004-01-01
In this paper, we first introduce the model of discrete-time neural networks with generalized input-output function and present a proof of the existence of a fixed point by Schauder fixed-point principle. Secondly, we study the uniformly asymptotical stability of equilibrium in non-autonomous discrete-time neural networks and give some sufficient conditions that guarantee the stability of it by using the converse theorem of Lyapunov function. Finally, several examples and numerical simulations are given to illustrate and reinforce our theories.
Short-Term Forecasting of Urban Water Consumption Based on the Largest Lyapunov Exponent
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
An approach for short-term forecasting of municipal water consumption was presented based on the largest Lyapunov exponent of chaos theory. The chaotic characteristics of time series of urban water consumption were examined by means of the largest Lyapunov exponent and correlation dimension. By using the largest Lyapunov exponent a short-term forecasting model for urban water consumption was developed, which was compared with the artificial neural network (ANN) approach in a case study. The result indicates that the model based on the largest Lyapunov exponent has higher prediction precision and forecasting stability than the ANN method, and its forecasting mean relative error is 9.6% within its maximum predictable time scale while it is 60.6% beyond the scale.
Application of local Lyapunov exponents to maneuver design and navigation in the three-body problem
Anderson, Rodney L.; Lo, Martin W.; Born, George H.
2003-01-01
Dynamical systems theory has recently been employed to design trajectories within the three-body problem for several missions. This research has applied one stability technique, the calculation of local Lyapunov exponents, to such trajectories. Local Lyapunov exponents give an indication of the effects that perturbations or maneuvers will have on trajectories over a specified time. A numerical comparison of local Lyapunov exponents was first made with the distance random perturbations traveled from a nominal trajectory, and the local Lyapunov exponents were found to correspond well with the perturbations that caused the greatest deviation from the nominal. This would allow them to be used as an indicator of the points where it would be important to reduce navigation uncertainties.
Application of local Lyapunov exponents to maneuver design and navigation in the three-body problem
Anderson, Rodney L.; Lo, Martin W.; Born, George H.
2003-01-01
Dynamical systems theory has recently been employed to design trajectories within the three-body problem for several missions. This research has applied one stability technique, the calculation of local Lyapunov exponents, to such trajectories. Local Lyapunov exponents give an indication of the effects that perturbations or maneuvers will have on trajectories over a specified time. A numerical comparison of local Lyapunov exponents was first made with the distance random perturbations traveled from a nominal trajectory, and the local Lyapunov exponents were found to correspond well with the perturbations that caused the greatest deviation from the nominal. This would allow them to be used as an indicator of the points where it would be important to reduce navigation uncertainties.
Lyapunov instabilities of Lennard-Jones fluids.
Yang, Hong-liu; Radons, Günter
2005-03-01
Recent work on many-particle systems reveals the existence of regular collective perturbations corresponding to the smallest positive Lyapunov exponents (LEs), called hydrodynamic Lyapunov modes. Until now, however, these modes have been found only for hard-core systems. Here we report results on Lyapunov spectra and Lyapunov vectors (LVs) for Lennard-Jones fluids. By considering the Fourier transform of the coordinate fluctuation density u((alpha)) (x,t) , it is found that the LVs with lambda approximately equal to 0 are highly dominated by a few components with low wave numbers. These numerical results provide strong evidence that hydrodynamic Lyapunov modes do exist in soft-potential systems, although the collective Lyapunov modes are more vague than in hard-core systems. In studying the density and temperature dependence of these modes, it is found that, when the value of the Lyapunov exponent lambda((alpha)) is plotted as function of the dominant wave number k(max) of the corresponding LV, all data from simulations with different densities and temperatures collapse onto a single curve. This shows that the dispersion relation lambda((alpha)) vs k(max) for hydrodynamical Lyapunov modes appears to be universal for the low-density cases studied here. Despite the wavelike character of the LVs, no steplike structure exists in the Lyapunov spectrum of the systems studied here, in contrast to the hard-core case. Further numerical simulations show that the finite-time LEs fluctuate strongly. We have also investigated localization features of LVs and propose a length scale to characterize the Hamiltonian spatiotemporal chaotic states.
Farivar, Faezeh; Shoorehdeli, Mahdi Aliyari
2012-01-01
In this paper, fault tolerant synchronization of chaotic gyroscope systems versus external disturbances via Lyapunov rule-based fuzzy control is investigated. Taking the general nature of faults in the slave system into account, a new synchronization scheme, namely, fault tolerant synchronization, is proposed, by which the synchronization can be achieved no matter whether the faults and disturbances occur or not. By making use of a slave observer and a Lyapunov rule-based fuzzy control, fault tolerant synchronization can be achieved. Two techniques are considered as control methods: classic Lyapunov-based control and Lyapunov rule-based fuzzy control. On the basis of Lyapunov stability theory and fuzzy rules, the nonlinear controller and some generic sufficient conditions for global asymptotic synchronization are obtained. The fuzzy rules are directly constructed subject to a common Lyapunov function such that the error dynamics of two identical chaotic motions of symmetric gyros satisfy stability in the Lyapunov sense. Two proposed methods are compared. The Lyapunov rule-based fuzzy control can compensate for the actuator faults and disturbances occurring in the slave system. Numerical simulation results demonstrate the validity and feasibility of the proposed method for fault tolerant synchronization.
Lyapunov analysis: from dynamical systems theory to applications
Cencini, Massimo; Ginelli, Francesco
2013-06-01
The study of deterministic laws of evolution has characterized the development of science since Newton's times. Chaos, namely the manifestation of irregular and unpredictable dynamics (not random but look random [1]), entered the debate on determinism at the end of the 19th century with the discovery of sensitivity to initial conditions, meaning that small infinitesimal differences in the initial state might lead to dramatic differences at later times. Poincaré [2, 3] was the first to realize that solutions of the three-body problem are generically highly sensitive to initial conditions. At about the same time, this property was recognized in geodesic flows with negative curvature by Hadamard [4]. One of the first experimental observations of chaos, as understood much later, was when irregular noise was heard by Van der Pol in 1927 [5] while studying a periodically forced nonlinear oscillator. Nevertheless, it was only with the advent of digital computing that chaos started to attract the interest of the wider scientific community. After the pioneering investigation of ergodicity in a chain of nonlinear oscillators by Fermi, Pasta and Ulam in 1955 [6], it was in the early 1960s that the numerical studies of Lorenz [7] and Hénon and Heiles [8] revealed that irregular and unpredictable motions are a generic feature of low-dimensional nonlinear deterministic systems. The existence and onset of chaos was then rigorously analyzed in several systems. While an exhaustive list of such mathematical proofs is beyond the scope of this preface, one should mention the contributions of Kolmogorov [9, 10], Chirikov [11], Smale [12], Ruelle and Takens [13], Li and Yorke [14] and Feigenbaum [15]. The characteristic Lyapunov exponents introduced by Oseledets in 1968 [16] are the fundamental quantities for measuring the sensitivity to initial conditions. Oseledets' work generalized the concept of Lyapunov stability to irregular trajectories building upon earlier studies of Birkhoff
Lagrange Theorem for polygroups
Directory of Open Access Journals (Sweden)
alireza sedighi
2014-12-01
Full Text Available So far?, ?isomorphism theorems in hyperstructure were proved for different structures of polygroups?, ?hyperrings and etc?. ?In this paper?, ?the polygroups properties is studied with the introduction of a suitable equivalence relation?. ?We show that the above relation is strongly regular?. ?Our main purpose in the paper is investigating Lagrang theorem and other expressing of isomorphism theorems for polygroups?.
Stabilizing model predictive control for constrained nonlinear distributed delay systems.
Mahboobi Esfanjani, R; Nikravesh, S K Y
2011-04-01
In this paper, a model predictive control scheme with guaranteed closed-loop asymptotic stability is proposed for a class of constrained nonlinear time-delay systems with discrete and distributed delays. A suitable terminal cost functional and also an appropriate terminal region are utilized to achieve asymptotic stability. To determine the terminal cost, a locally asymptotically stabilizing controller is designed and an appropriate Lyapunov-Krasoskii functional of the locally stabilized system is employed as the terminal cost. Furthermore, an invariant set for locally stabilized system which is established by using the Razumikhin Theorem is used as the terminal region. Simple conditions are derived to obtain terminal cost and terminal region in terms of Bilinear Matrix Inequalities. The method is illustrated by a numerical example.
A Converse Sum of Squares Lyapunov Result with a Degree Bound
Peet, Matthew M
2012-01-01
Sum of Squares programming has been used extensively over the past decade for the stability analysis of nonlinear systems but several questions remain unanswered. In this paper, we show that exponential stability of a polynomial vector field on a bounded set implies the existence of a Lyapunov function which is a sum-of-squares of polynomials. In particular, the main result states that if a system is exponentially stable on a bounded nonempty set, then there exists an SOS Lyapunov function which is exponentially decreasing on that bounded set. The proof is constructive and uses the Picard iteration. A bound on the degree of this converse Lyapunov function is also given. This result implies that semidefinite programming can be used to answer the question of stability of a polynomial vector field with a bound on complexity.
Invariance of Lyapunov exponents and Lyapunov dimension for regular and irregular linearizations
Kuznetsov, N. V.; Alexeeva, T. A.; Leonov, G. A.
2014-01-01
Nowadays the Lyapunov exponents and Lyapunov dimension have become so widespread and common that they are often used without references to the rigorous definitions or pioneering works. It may lead to a confusion since there are at least two well-known definitions, which are used in computations: the upper bounds of the exponential growth rate of the norms of linearized system solutions (Lyapunov characteristic exponents, LCEs) and the upper bounds of the exponential growth rate of the singula...
Covariant Lyapunov vectors for rigid disk systems.
Bosetti, Hadrien; Posch, Harald A
2010-10-05
We carry out extensive computer simulations to study the Lyapunov instability of a two-dimensional hard-disk system in a rectangular box with periodic boundary conditions. The system is large enough to allow the formation of Lyapunov modes parallel to the x-axis of the box. The Oseledec splitting into covariant subspaces of the tangent space is considered by computing the full set of covariant perturbation vectors co-moving with the flow in tangent space. These vectors are shown to be transversal, but generally not orthogonal to each other. Only the angle between covariant vectors associated with immediate adjacent Lyapunov exponents in the Lyapunov spectrum may become small, but the probability of this angle to vanish approaches zero. The stable and unstable manifolds are transverse to each other and the system is hyperbolic.
Infinitesimal Lyapunov functions and singular-hyperbolicity
Araujo, Vitor
2012-01-01
We present an extension of the notion of infinitesimal Lyapunov function to singular flows on three-dimensional manifolds, and show how this technique provides a characterization of partially hyperbolic structures for invariant sets for such flows, and also of singular-hyperbolicity. In the absence of singularities, we can also rephrase uniform hyperbolicity with the language of infinitesimal Lyapunov functions. These conditions are expressed using the vector field X and its space derivative DX together with an infinitesimal Lyapunov function only and are reduced to checking that a certain symmetric operator is positive definite on the trapping region: we show how to express partial hyperbolicity using only the interplay between the infinitesimal generator X of the flow X_t, its derivative DX and the infinitesimal Lyapunov function.
Levinson, N
1940-01-01
A typical gap theorem of the type discussed in the book deals with a set of exponential functions { \\{e^{{{i\\lambda}_n} x}\\} } on an interval of the real line and explores the conditions under which this set generates the entire L_2 space on this interval. A typical gap theorem deals with functions f on the real line such that many Fourier coefficients of f vanish. The main goal of this book is to investigate relations between density and gap theorems and to study various cases where these theorems hold. The author also shows that density- and gap-type theorems are related to various propertie
Two No-Go Theorems on Superconductivity
Tada, Yasuhiro
2016-01-01
We study lattice superconductors such as attractive Hubbard models. As is well known, Bloch's theorem asserts absence of persistent current in ground states and equilibrium states for general fermion systems. While the statement of the theorem is true, we can show that the theorem cannot exclude possibility of a surface persistent current. Such a current can be stabilized by boundary magnetic fields which do not penetrate into the bulk region of a superconductor, provided emergence of massive photons, i.e., Meissner effect. Therefore, we can expect that a surface persistent current is realized for a ground/equilibrium state in the sense of stability against local perturbations. We also apply Elitzur's theorem to superconductors at finite temperatures. As a result, we prove absence of symmetry breaking of the global U(1) phase of electrons for almost all gauge fixings. These observations suggest that the nature of superconductivity is the emergence of massive photons rather than the symmetry breaking of the U(...
Short-time Lyapunov exponent analysis
Vastano, J. A.
1990-01-01
A new technique for analyzing complicated fluid flows in numerical simulations has been successfully tested. The analysis uses short time Lyapunov exponent contributions and the associated Lyapunov perturbation fields. A direct simulation of the Taylor-Couette flow just past the onset of chaos demonstrated that this new technique marks important times during the system evolution and identifies the important flow features at those times. This new technique will now be applied to a 'minimal' turbulent channel.
Short-time Lyapunov exponent analysis
Vastano, J. A.
1990-01-01
A new technique for analyzing complicated fluid flows in numerical simulations has been successfully tested. The analysis uses short time Lyapunov exponent contributions and the associated Lyapunov perturbation fields. A direct simulation of the Taylor-Couette flow just past the onset of chaos demonstrated that this new technique marks important times during the system evolution and identifies the important flow features at those times. This new technique will now be applied to a 'minimal' turbulent channel.
一类线性脉冲微分系统的变差稳定性%Variational Stability for a Class of Linear Differential Systems with Impulses
Institute of Scientific and Technical Information of China (English)
李宝麟; 王倩倩
2012-01-01
利用Henstock积分、Lyapunov函数以及脉冲微分系统理论,讨论了一类带脉冲效应的线性微分系统有界变差解的稳定性,并建立了有界变差解的变差稳定性和渐近变差稳定性的Lyapunov型定理.%By using the Henstock integral, Lyapunov function and the theory of impulsive differential system , the variational stability of bounded variation solutions for a class of linear differential systems with impulses was discussed, and trie Lyapunov type theorems for variational stability and asymptotically variational stability of bounded variation solutions were established.
Stability and response bounds of non-conservative linear systems
DEFF Research Database (Denmark)
Pommer, Christian
2003-01-01
For a linear system of second order differential equations the stability is studied by Lyapunov's direct method. The Lyapunov matrix equation is solved and a sufficient condition for stability is expressed by the system matrices. For a system which satisfies the condition for stability the Lyapunov...
Circular orbit spacecraft control at the L4 point using Lyapunov functions
Agrawal, Rachana
2015-01-01
The objective of this work is to demonstrate the utility of Lyapunov functions in control synthesis for the purpose of maintaining and stabilizing a spacecraft in a circular orbit around the L4 point in the circular restricted three body problem (CRTBP). Incorporating the requirements of a fixed radius orbit and a desired angular momentum, a Lyapunov function is constructed and the requisite analysis is performed to obtain a controller. Asymptotic stability is proved in a defined region around the L4 point using LaSalle's principle.
Lyapunov function and the basin of attraction for a single-joint muscle-skeletal model.
Giesl, Peter; Wagner, Heiko
2007-04-01
This paper provides an explicit Lyapunov function for a general single-joint muscle-skeletal model. Using this Lyapunov function one can determine analytically large subsets of the basin of attraction of an asymptotically stable equilibrium. Besides providing an analytical tool for the analysis of such a system we consider an elbow model and show that the theoretical predictions are in agreement with experimental results. Moreover, we can thus distinguish between regions where the self-stabilizing properties of the muscle-skeletal system guarantee stability and regions where nerval control and reflexes are necessary.
关于李雅普诺夫函数的几点注记%Some notes on Lyapunov function
Institute of Scientific and Technical Information of China (English)
徐助跃
2013-01-01
Three definitions and three theorems of Lyapunov function were put forward, and then these definitions and theorems were illustrated with some examples. Finally, under the generalized Hurwitz condition, the "three-step method" was put forward to construct Lyapunov function of nonlinear system.% 给出了关于李雅普诺夫函数的3个定义和3个定理，举例说明了所得定义和定理的应用，并给出了利用广义霍维茨条件构造非线性系统李雅普诺夫函数的“三步法”。
Stability and Stabilization of Networked Control System with Forward and Backward Random Time Delays
Directory of Open Access Journals (Sweden)
Ye-Guo Sun
2012-01-01
Full Text Available This paper deals with the problem of stabilization for a class of networked control systems (NCSs with random time delay via the state feedback control. Both sensor-to-controller and controller-to-actuator delays are modeled as Markov processes, and the resulting closed-loop system is modeled as a Markovian jump linear system (MJLS. Based on Lyapunov stability theorem combined with Razumikhin-based technique, a new delay-dependent stochastic stability criterion in terms of bilinear matrix inequalities (BMIs for the system is derived. A state feedback controller that makes the closed-loop system stochastically stable is designed, which can be solved by the proposed algorithm. Simulations are included to demonstrate the theoretical result.
Lyapunov Orbits in the Jupiter System Using Electrodynamic Tethers
Bokelmann, Kevin; Russell, Ryan P.; Lantoine, Gregory
2013-01-01
Various researchers have proposed the use of electrodynamic tethers for power generation and capture from interplanetary transfers. The effect of tether forces on periodic orbits in Jupiter-satellite systems are investigated. A perturbation force is added to the restricted three-body problem model and a series of simplifications allows development of a conservative system that retains the Jacobi integral. Expressions are developed to find modified locations of equilibrium positions. Modified families of Lyapunov orbits are generated as functions of tether size and Jacobi integral. Zero velocity curves and stability analyses are used to evaluate the dynamical properties of tether-modified orbits.
Lyapunov Orbits in the Jupiter System Using Electrodynamic Tethers
Bokelmann, Kevin; Russell, Ryan P.; Lantoine, Gregory
2013-01-01
Various researchers have proposed the use of electrodynamic tethers for power generation and capture from interplanetary transfers. The effect of tether forces on periodic orbits in Jupiter-satellite systems are investigated. A perturbation force is added to the restricted three-body problem model and a series of simplifications allows development of a conservative system that retains the Jacobi integral. Expressions are developed to find modified locations of equilibrium positions. Modified families of Lyapunov orbits are generated as functions of tether size and Jacobi integral. Zero velocity curves and stability analyses are used to evaluate the dynamical properties of tether-modified orbits.
Design of Connectivity Preserving Flocking Using Control Lyapunov Function
Directory of Open Access Journals (Sweden)
Bayu Erfianto
2016-01-01
Full Text Available This paper investigates cooperative flocking control design with connectivity preserving mechanism. During flocking, interagent distance is measured to determine communication topology of the flocks. Then, cooperative flocking motion is built based on cooperative artificial potential field with connectivity preserving mechanism to achieve the common flocking objective. The flocking control input is then obtained by deriving cooperative artificial potential field using control Lyapunov function. As a result, we prove that our flocking protocol establishes group stabilization and the communication topology of multiagent flocking is always connected.
The asymmetric sandwich theorem
Simons, Stephen
2011-01-01
We discuss the asymmetric sandwich theorem, a generalization of the Hahn-Banach theorem. As applications, we derive various results on the existence of linear functionals that include bivariate, trivariate and quadrivariate generalizations of the Fenchel duality theorem. Most of the results are about affine functions defined on convex subsets of vector spaces, rather than linear functions defined on vector spaces. We consider both results that use a simple boundedness hypothesis (as in Rockafellar's version of the Fenchel duality theorem) and also results that use Baire's theorem (as in the Robinson-Attouch-Brezis version of the Fenchel duality theorem). This paper also contains some new results about metrizable topological vector spaces that are not necessarily locally convex.
DEFF Research Database (Denmark)
Ribard, Nicolas; Wisniewski, Rafael; Sloth, Christoffer
2016-01-01
In the paper, we strive to develop an algorithm that simultaneously computes a polynomial control and a polynomial Lyapunov function. This ensures asymptotic stability of the designed feedback system. The above problem is translated to a certificate of positivity. To this end, we use the represen......In the paper, we strive to develop an algorithm that simultaneously computes a polynomial control and a polynomial Lyapunov function. This ensures asymptotic stability of the designed feedback system. The above problem is translated to a certificate of positivity. To this end, we use...... the representation of the given control system in Bernstein basis. Subsequently, the control synthesis problem is reduced to finite number of evaluations of a polynomial on vertices of cubes in the space of parameters representing admissible controls and Lyapunov functions....
Absolute Stability for Lurie Control System with Unbound Time Delays
Institute of Scientific and Technical Information of China (English)
王天成; 王耀才; 洪留荣
2004-01-01
Time delay existes widely in various real engineering systems and can result in unsatisfactory performance or even an instability of control systems. Therefore, to investigate the stability for time delay systems is of vitul importance in control theory and its applications. Many researchers have studied the stability criteria of systems with constant delay or bound varying time delay, but few of them studied large time delay or unbound time delay. Large time delay existes commonly in various engineering applications. In this paper, the absolute stability of Lurie type direct control systems and indirect control systems with several time delays are discussed. Based on Lyapunov theory, the new delay dependent absolute stability criteria are derived. In our theorem, time delays can be unbound functions, which shows that the results are less conservative than that of existed criteria.
Institute of Scientific and Technical Information of China (English)
王静; 蒋刚
2013-01-01
针对轮式移动机器人在实际运行中受环境因数影响的情况,采用扩展卡尔曼滤波(EKF)算法融合里程计与超声波的观测数据,对机器人的参考轨迹信息进行校正.在机器人动力学模型的基础上,运用Lyapunov直接法,构造具有全局渐近稳定的跟踪控制器,对机器人进行轨迹跟踪.根据Lyapunov稳定性定理证明了系统的全局稳定性.仿真结果表明,数据滤波与Lyapunov方法结合的跟踪控制器效果良好.%According to the fact that the wheeled mobile robots are influenced by the environmental factor in practice,the information of reference trajectory of the robot was corrected by using the extended Kalman filter (EKF) algorithm fusion odometry and ultrasonic observation data.Based on the robot dynamic model,a global asymptotical stable tracking controller was constructed by using the Lyapunov direct method,and the global stability of the system was proved by using Lyapunov stability theorem.The simulation results of this paper showed that the tracking controller,which combined both the data filtering and Lyapunov method,has better efficiency.
A unified approach to finite-time hyperbolicity which extends finite-time Lyapunov exponents
Doan, T. S.; Karrasch, D.; Nguyen, T. Y.; Siegmund, S.
A hyperbolicity notion for linear differential equations x˙=A(t)x, t∈[t-,t+], is defined which unifies different existing notions like finite-time Lyapunov exponents (Haller, 2001, [13], Shadden et al., 2005, [24]), uniform or M-hyperbolicity (Haller, 2001, [13], Berger et al., 2009, [6]) and (t-,(t+-t-))-dichotomy (Rasmussen, 2010, [21]). Its relation to the dichotomy spectrum (Sacker and Sell, 1978, [23], Siegmund, 2002, [26]), D-hyperbolicity (Berger et al., 2009, [6]) and real parts of the eigenvalues (in case A is constant) is described. We prove a spectral theorem and provide an approximation result for the spectral intervals.
A survey of quantum Lyapunov control methods.
Cong, Shuang; Meng, Fangfang
2013-01-01
The condition of a quantum Lyapunov-based control which can be well used in a closed quantum system is that the method can make the system convergent but not just stable. In the convergence study of the quantum Lyapunov control, two situations are classified: nondegenerate cases and degenerate cases. For these two situations, respectively, in this paper the target state is divided into four categories: the eigenstate, the mixed state which commutes with the internal Hamiltonian, the superposition state, and the mixed state which does not commute with the internal Hamiltonian. For these four categories, the quantum Lyapunov control methods for the closed quantum systems are summarized and analyzed. Particularly, the convergence of the control system to the different target states is reviewed, and how to make the convergence conditions be satisfied is summarized and analyzed.
Lyapunov exponents computation for hybrid neurons.
Bizzarri, Federico; Brambilla, Angelo; Gajani, Giancarlo Storti
2013-10-01
Lyapunov exponents are a basic and powerful tool to characterise the long-term behaviour of dynamical systems. The computation of Lyapunov exponents for continuous time dynamical systems is straightforward whenever they are ruled by vector fields that are sufficiently smooth to admit a variational model. Hybrid neurons do not belong to this wide class of systems since they are intrinsically non-smooth owing to the impact and sometimes switching model used to describe the integrate-and-fire (I&F) mechanism. In this paper we show how a variational model can be defined also for this class of neurons by resorting to saltation matrices. This extension allows the computation of Lyapunov exponent spectrum of hybrid neurons and of networks made up of them through a standard numerical approach even in the case of neurons firing synchronously.
Stability of linear switched systems with quadratic bounds and Observability of bilinear systems
Balde, Moussa
2012-01-01
The aim of this paper is to give sufficient conditions for a switched linear system defined by a pair of Hurwitz matrices that share a common but not strict quadratic Lyapunov function to be GUAS. We show that this property is equivalent to the uniform observability of a bilinear system defined on a subspace whose dimension is in most cases much smaller than the dimension of the switched system. Some sufficient conditions of uniform asymptotic stability are then deduced from the equivalence theorem, and illustrated by examples.
Directory of Open Access Journals (Sweden)
Wu Huaiqin
2009-01-01
Full Text Available This paper considers a new class of additive neural networks where the neuron activations are modelled by discontinuous functions with nonlinear growth. By Leray-Schauder alternative theorem in differential inclusion theory, matrix theory, and generalized Lyapunov approach, a general result is derived which ensures the existence and global asymptotical stability of a unique periodic solution for such neural networks. The obtained results can be applied to neural networks with a broad range of activation functions assuming neither boundedness nor monotonicity, and also show that Forti's conjecture for discontinuous neural networks with nonlinear growth activations is true.
Time-Delay Systems Lyapunov Functionals and Matrices
Kharitonov, Vladimir L
2013-01-01
Stability is one of the most studied issues in the theory of time-delay systems, but the corresponding chapters of published volumes on time-delay systems do not include a comprehensive study of a counterpart of classical Lyapunov theory for linear delay free systems. The principal goal of the book is to fill this gap, and to provide readers with a systematic and exhaustive treatment of the basic concepts of the Lyapunov-Krasovskii approach to the stability analysis of linear time-delay systems. The book is organized into two parts. The first part is dedicated to the case of retarded type time-delay systems; it consists of four chapters, which respectively deal with results concerning the existence and uniqueness of the solutions of an initial value problem, the class of linear systems with one delay, the case of systems with several delays, and the case of systems with distributed delays. The second part of the book studies the case of neutral type time-delay systems, containing three chapters that e...
A Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel.
Abdeljawad, Thabet
2017-01-01
In this article, we extend fractional operators with nonsingular Mittag-Leffler kernels, a study initiated recently by Atangana and Baleanu, from order [Formula: see text] to higher arbitrary order and we formulate their correspondent integral operators. We prove existence and uniqueness theorems for the Caputo ([Formula: see text]) and Riemann ([Formula: see text]) type initial value problems by using the Banach contraction theorem. Then we prove a Lyapunov type inequality for the Riemann type fractional boundary value problems of order [Formula: see text] in the frame of Mittag-Leffler kernels. Illustrative examples are analyzed and an application as regards the Sturm-Liouville eigenvalue problem in the sense of this fractional calculus is given as well.
Mizuta, Keisuke; Tokita, Takashi; Ito, Yatsuji; Aoki, Mitsuhiro; Kuze, Bunya
2009-12-01
largest Lyapunov exponents has a different significance from instability of standing posture indicated by a conventional analysis. We propose that the largest Lyapunov exponents may be an useful subsidiary measure to evaluate postural stability and its change due to vestibular dysfunction.
2011-01-01
Arvanitakis established recently a theorem which is a common generalization of Michael's convex selection theorem and Dugundji's extension theorem. In this note we provide a short proof of a more general version of Arvanitakis' result.
The Lyapunov dimension and its estimation via the Leonov method
Energy Technology Data Exchange (ETDEWEB)
Kuznetsov, N.V., E-mail: nkuznetsov239@gmail.com
2016-06-03
Highlights: • Survey on effective analytical approach for Lyapunov dimension estimation, proposed by Leonov, is presented. • Invariance of Lyapunov dimension under diffeomorphisms and its connection with Leonov method are demonstrated. • For discrete-time dynamical systems an analog of Leonov method is suggested. - Abstract: Along with widely used numerical methods for estimating and computing the Lyapunov dimension there is an effective analytical approach, proposed by G.A. Leonov in 1991. The Leonov method is based on the direct Lyapunov method with special Lyapunov-like functions. The advantage of the method is that it allows one to estimate the Lyapunov dimension of invariant sets without localization of the set in the phase space and, in many cases, to get effectively an exact Lyapunov dimension formula. In this work the invariance of the Lyapunov dimension with respect to diffeomorphisms and its connection with the Leonov method are discussed. For discrete-time dynamical systems an analog of Leonov method is suggested. In a simple but rigorous way, here it is presented the connection between the Leonov method and the key related works: Kaplan and Yorke (the concept of the Lyapunov dimension, 1979), Douady and Oesterlé (upper bounds of the Hausdorff dimension via the Lyapunov dimension of maps, 1980), Constantin, Eden, Foiaş, and Temam (upper bounds of the Hausdorff dimension via the Lyapunov exponents and Lyapunov dimension of dynamical systems, 1985–90), and the numerical calculation of the Lyapunov exponents and dimension.
Stabilisation of a class of 2-DOF underactuated mechanical systems via direct Lyapunov approach
Turker, Turker; Gorgun, Haluk; Cansever, Galip
2013-06-01
This paper represents an alternative stabilisation procedure for a class of two degree-of-freedom underactuated mechanical systems based on a set of transformations and a Lyapunov function. After simplifying dynamic equations of the system via partial feedback linearisation and coordinate changes, the stability of the system is provided with Lyapunov's direct method. Proposed control scheme is used on two different examples and asymptotic convergence for each system is proven by means of La Salle's invariance principle. The designed controller is successfully illustrated through numerical simulations for each example.
Institute of Scientific and Technical Information of China (English)
冯玉瑚; 朱凡昌
2004-01-01
Solutions of fuzzy differential equations provide a noteworthy example of time-dependent fuzzy sets. The purpose of this paper is to introduce functions of a suitable Lyapunov-like type and to show the existence and uniqueness theorem for the Cauchy problem of fuzzy differential equations under non-Lipschitz conditions. The comparison principles and the existence and uniqueness theorems of this paper generalize many well-known results up to now.
Lyapunov Function Synthesis - Infeasibility and Farkas' Lemma
DEFF Research Database (Denmark)
Leth, Tobias; Wisniewski, Rafal; Sloth, Christoffer
2017-01-01
In this paper we prove the convergence of an algorithm synthesising continuous piecewise-polynomial Lyapunov functions for polynomial vector elds dened on simplices. We subsequently modify the algorithm to sub-divide locally by utilizing information from infeasible linear problems. We prove...
Controllability of semilinear matrix Lyapunov systems
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Bhaskar Dubey
2013-02-01
Full Text Available In this article, we establish some sufficient conditions for the complete controllability of semilinear matrix Lyapunov systems involving Lipschitzian and non-Lipschitzian nonlinearities. In case of non-Lipschitzian nonlinearities, we assume that nonlinearities are of monotone type.
Lyapunov exponents for continuous random transformations
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
In this paper, the concept of Lyapunov exponent is generalized to random transformations that are not necessarily differentiable. For a class of random repellers and of random hyperbolic sets obtained via small perturbations of deterministic ones respectively, the new exponents are shown to coincide with the classical ones.
Stability of simultaneously triangularizable switched systems on hybrid domains
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Geoffrey Eisenbarth
2014-03-01
Full Text Available In this paper, we extend the results of [8, 15, 22], which provide sufficient conditions for the global exponential stability of switched systems under arbitrary switching via the existence of a common quadratic Lyapunov function. In particular, we extend the Lie algebraic results in [15] to switched systems with hybrid non-uniform discrete and continuous domains, a direct unifying generalization of switched systems on R and Z, and extend the results in [8, 22] to a larger class of switched systems, namely those whose subsystem matrices are simultaneously triangularizable. In addition, we explore an easily checkable characterization of our required hypotheses for the theorems. Finally, conditions are provided under which there exists a stabilizing switching pattern for a collection of (not necessarily stable linear systems that are simultaneously triangularizable and separate criteria are formed which imply the stability of the system under a given switching pattern given a priori.
Canada, Antonio
2011-01-01
Several different problems make the study of the so called Lyapunov type inequalities of great interest, both in pure and applied mathematics. Although the original historical motivation was the study of the stability properties of the Hill equation (which applies to many problems in physics and engineering), other questions that arise in systems at resonance, crystallography, isoperimetric problems, Rayleigh type quotients, etc. lead to the study of $L_p$ Lyapunov inequalities ($1\\leq p\\leq \\infty$) for differential equations. In this work we review some recent results on these kinds of questions which can be formulated as optimal control problems. In the case of Ordinary Differential Equations, we consider periodic and antiperiodic boundary conditions at higher eigenvalues and by using a more accurate version of the Sturm separation theory, an explicit optimal result is obtained. Then, we establish Lyapunov inequalities for systems of equations. To this respect, a key point is the characterization of the be...
GA and Lyapunov theory-based hybrid adaptive fuzzy controller for non-linear systems
Roy, Ananya; Das Sharma, Kaushik
2015-02-01
In this present article, a new hybrid methodology for designing stable adaptive fuzzy logic controllers (AFLCs) for a class of non-linear system is proposed. The proposed design strategy exploits the features of genetic algorithm (GA)-based stochastic evolutionary global search technique and Lyapunov theory-based local adaptation scheme. The objective is to develop a methodology for designing AFLCs with optimised free parameters and guaranteed closed-loop stability. Simultaneously, the proposed method introduces automation in the design process. The stand-alone Lyapunov theory-based design, GA-based design and proposed hybrid GA-Lyapunov design methodologies are implemented for two benchmark non-linear plants in simulation case studies with different reference signals and one experimental case study. The results demonstrate that the hybrid design methodology outperforms the other control strategies on the whole.
Lyapunov-based boundary feedback control in multi-reach canals
Institute of Scientific and Technical Information of China (English)
CEN LiHui; XI YuGeng
2009-01-01
This paper presents a Lyapunov-based approach to design the boundary feedback control for an open-channel network composed of a cascade of multi-reach canals, each described by a pair of Saint-Venant equations. The weighted sum of entropies of the multi-reaches is adopted to construct the Lyapunov function. The time derivative of the Lyapunov function is expressed by the water depth variations at the gate boundaries, based on which a class of boundary feedback controllers is presented to guarantee the local asymptotic closed-loop stability. The advantage of this approach is that only the water level depths at the gate boundaries are measured as the feedback.
The Multivariate Largest Lyapunov Exponent as an Age-Related Metric of Quiet Standing Balance
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Kun Liu
2015-01-01
Full Text Available The largest Lyapunov exponent has been researched as a metric of the balance ability during human quiet standing. However, the sensitivity and accuracy of this measurement method are not good enough for clinical use. The present research proposes a metric of the human body’s standing balance ability based on the multivariate largest Lyapunov exponent which can quantify the human standing balance. The dynamic multivariate time series of ankle, knee, and hip were measured by multiple electrical goniometers. Thirty-six normal people of different ages participated in the test. With acquired data, the multivariate largest Lyapunov exponent was calculated. Finally, the results of the proposed approach were analysed and compared with the traditional method, for which the largest Lyapunov exponent and power spectral density from the centre of pressure were also calculated. The following conclusions can be obtained. The multivariate largest Lyapunov exponent has a higher degree of differentiation in differentiating balance in eyes-closed conditions. The MLLE value reflects the overall coordination between multisegment movements. Individuals of different ages can be distinguished by their MLLE values. The standing stability of human is reduced with the increment of age.
The Multivariate Largest Lyapunov Exponent as an Age-Related Metric of Quiet Standing Balance.
Liu, Kun; Wang, Hongrui; Xiao, Jinzhuang
2015-01-01
The largest Lyapunov exponent has been researched as a metric of the balance ability during human quiet standing. However, the sensitivity and accuracy of this measurement method are not good enough for clinical use. The present research proposes a metric of the human body's standing balance ability based on the multivariate largest Lyapunov exponent which can quantify the human standing balance. The dynamic multivariate time series of ankle, knee, and hip were measured by multiple electrical goniometers. Thirty-six normal people of different ages participated in the test. With acquired data, the multivariate largest Lyapunov exponent was calculated. Finally, the results of the proposed approach were analysed and compared with the traditional method, for which the largest Lyapunov exponent and power spectral density from the centre of pressure were also calculated. The following conclusions can be obtained. The multivariate largest Lyapunov exponent has a higher degree of differentiation in differentiating balance in eyes-closed conditions. The MLLE value reflects the overall coordination between multisegment movements. Individuals of different ages can be distinguished by their MLLE values. The standing stability of human is reduced with the increment of age.
A Hamiltonian KAM theorem for bundles of Lagrangean tori
Broer, HW; Cushman, RH; Fasso, F; Dumortier, F; Broer, H; Mawhin, J; Vanderbauwhede, A; Lunel, SV
2005-01-01
The classical KAM theorem deals with Lagrangean invariant tori in nearly integrable Hamiltonian systems. The stability formulation of the KAM theorem states that, when restricting to a large measure Diophantine "Cantor set" of such tori, the integrable approximation is smoothly conjugate to the near
Directory of Open Access Journals (Sweden)
M. Widi Triyatno
2015-03-01
Full Text Available Disturbances in the operation of the power system may cause disturbance in voltage stability. Therefore, dynamic voltage stability analysis before and after disturbance needs to be performed. This paper proposes dynamic voltage stability prediction using maximum Lyapunov exponent with Lampung’s electrical system as case study. Voltage stability simulation is performed with various types of disturbances that occur at line between of Baturaja substation and Bukit Kemuning substation. Time-series data of voltage measurement of simulation results at GI Baturaja is applied for voltage stability prediction analysis using maximum Lyapunov exponent. With the same number of data samples and the same time for circuit breakers to interrupt disturbances, the simulation results using maximum Lyapunov exponent show that the voltage can be stabilized at 1.7 seconds after the occurrence of the three-phase disturbance, at 1.2 seconds after the occurrence of the phase-to-ground disturbance, at 0,9 second after the occurrence of the disturbance between phase, at 1.2 seconds after the occurrence of the loss of line disturbance and 1.4 seconds after the occurrence of the loss of load disturbance. The amount of data samples used in analysis affect the time for the voltage reaches stability.
Directory of Open Access Journals (Sweden)
Ming-Chorng Hwang
2015-01-01
Full Text Available A theoretic formulation on how traffic time information distributed by ITS operations influences the trajectory of network flows is presented in this paper. The interactions between users and ITS operator are decomposed into three parts: (i travel time induced path flow dynamics (PFDTT; (ii demand induced path flow dynamics (PFDD; and (iii predicted travel time dynamics for an origin-destination (OD pair (PTTDOD. PFDTT describes the collective results of user’s daily route selection by pairwise comparison of path travel time provided by ITS services. The other two components, PTTDOD and PFDD, are concentrated on the evolutions of system variables which are predicted and observed, respectively, by ITS operators to act as a benchmark in guiding the target system towards an expected status faster. In addition to the delivered modelings, the stability theorem of the equilibrium solution in the sense of Lyapunov stability is also provided. A Lyapunov function is developed and employed to the proof of stability theorem to show the asymptotic behavior of the aimed system. The information of network flow dynamics plays a key role in traffic control policy-making. The evaluation of ITS-based strategies will not be reasonable without a well-established modeling of network flow evolutions.
Directory of Open Access Journals (Sweden)
Yi-You Hou
2014-01-01
Full Text Available This paper considers the problem of the robust stability for the nonlinear system with time-varying delay and parameters uncertainties. Based on the H∞ theorem, Lyapunov-Krasovskii theory, and linear matrix inequality (LMI optimization technique, the H∞ quasi-sliding mode controller and switching function are developed such that the nonlinear system is asymptotically stable in the quasi-sliding mode and satisfies the disturbance attenuation (H∞-norm performance. The effectiveness and accuracy of the proposed methods are shown in numerical simulations.
Direct adaptive control for nonlinear uncertain system based on control Lyapunov function method
Institute of Scientific and Technical Information of China (English)
Chen Yimei; Han Zhengzhi; Tang Houjun
2006-01-01
The problem of adaptive stabilization of a class of multi-input nonlinear systems with unknown parameters both in the state vector-field and the input vector-field has been considered. By employing the control Lyapunov function method, a direct adaptive controller is designed to complete the global adaptive stability of the uncertain system. At the same time, the controller is also verified to possess the optimality. Example and simulations are provided to illustrate the effectiveness of the proposed method.
Directory of Open Access Journals (Sweden)
Zhixiong Zhong
2013-01-01
Full Text Available The stability analysis and stabilization of Takagi-Sugeno (T-S fuzzy delta operator systems with time-varying delay are investigated via an input-output approach. A model transformation method is employed to approximate the time-varying delay. The original system is transformed into a feedback interconnection form which has a forward subsystem with constant delays and a feedback one with uncertainties. By applying the scaled small gain (SSG theorem to deal with this new system, and based on a Lyapunov Krasovskii functional (LKF in delta operator domain, less conservative stability analysis and stabilization conditions are obtained. Numerical examples are provided to illustrate the advantages of the proposed method.
On the Equivalence of Weyl Theorem and Generalized Weyl Theorem
Institute of Scientific and Technical Information of China (English)
M. BERKANI
2007-01-01
We know that an operator T acting on a Banach space satisfying generalized Weyl's theorem also satisfies Weyl's theo rem. Conversely we show that if all isolated eigenvalues of T are poles of its resolvent and if T satisfies Weyl's theorem, then it also satisfies generalized Weyl's theorem. We give also a similar result for the equivalence of a-Weyl's theorem and generalized a-Weyl's theorem. Using these results, we study the case of polaroid operators, and in particular paranormal operators.
To string together six theorems of physics by Pythagoras theorem
Cui, H Y
2002-01-01
In this paper, we point out that there are at lest six theorems in physics sharing common virtue of Pythagoras theorem, so that it is possible to string these theorems together with the Pythagoras theorem for physics teaching, the six theorems are Newton's three laws of motion, universal gravitational force, Coulomb's law, and the formula of relativistic dynamics. Knowing the internal relationships between them, which have never been clearly revealed by other author, will benefit the logic of physics teaching.
Trigonometry, Including Snell's Theorem.
Kent, David
1980-01-01
Aspects of the instruction of trigonometry in secondary school mathematics are reviewed. Portions of this document cover basic introductions, a student-developed theorem, the cosine rule, inverse functions, and a sample outdoor activity. (MP)
Trigonometry, Including Snell's Theorem.
Kent, David
1980-01-01
Aspects of the instruction of trigonometry in secondary school mathematics are reviewed. Portions of this document cover basic introductions, a student-developed theorem, the cosine rule, inverse functions, and a sample outdoor activity. (MP)
随机微分方程LaSalle定理%LaSalle's Theorem for Stochastic Differential Equations
Institute of Scientific and Technical Information of China (English)
黄庆道; 田萍; 王国明
2003-01-01
In this paper,we establish a the LaSalle's theorem for stochastic differential equation based on Li's work,and give a more general Lyapunov function which it is more easy to apply.Our work has partly generalized Mao's work.
Coincidence Point Theorems, Intersection Theorems and Saddle Point Theorems on FC-spaces
Institute of Scientific and Technical Information of China (English)
PIAO YONG-JIE; YIN ZHE
2009-01-01
In this paper, we first give the definitions of finitely continuous topological space and FC-subspace generated by some set, and obtain coincidence point theorem, whole intersection theorems and Ky Fan type matching theorems, and finally discuss the existence of saddle point as an application of coincidence point theorem.
Clustering and synchronization with positive Lyapunov exponents
Mendes, R V
1998-01-01
Clustering and correlation effects are frequently observed in chaotic systems in situations where, because of the positivity of the Lyapunov exponents, no dimension reduction is to be expected. In this paper, using a globally coupled network of Bernoulli units, one finds a general mechanism by which strong correlations and slow structures are obtained at the synchronization edge. A structure index is defined, which diverges at the transition points. Some conclusions are drawn concerning the construction of an ergodic theory of self-organization.
Control of acrobot based on Lyapunov function
Institute of Scientific and Technical Information of China (English)
赖旭芝; 吴敏; 佘锦华
2004-01-01
Fuzzy control based on Lyapunov function was employed to control the posture and the energy of an acrobot to make the transition from upswing control to balance control smoothly and stably. First, a control law based on Lyapunov function was used to control the angle and the angular velocity of the second link towards zero when the energy of the acrobot reaches the potential energy at the unstable straight-up equilibrium position in the upswing process. The controller based on Lyapunov function makes the second link straighten nature relatively to the first link. At the same time, a fuzzy controller was designed to regulate the parameters of the upper control law to keep the change of the energy of the acrobot to a minimum, so that the switching from upswing to balance can be properly carried out and the acrobot can enter the balance quickly. The results of simulation show that the switching from upswing to balance can be completed smoothly, and the control effect of the acrobot is improved greatly.
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Mohd Ariffanan Mohd Basri
2016-07-01
Full Text Available In this paper, the dynamic model of quadrotor helicopter has been mathematically formulated. Then, an intelligent backstepping controller (IBC is designed for the quadrotor altitude and attitude stabilization in the existence of external disturbances and measurement noise. The designed controller consists of a backstepping controller which can automatically select its parameters on-line by a fuzzy supervisory mechanism. The stability criterion for the stabilization of the quadrotor is proven by the Lyapunov theorem. Several numerical simulations using the dynamic model of a four degree of freedom (DOF quadrotor helicopter show the effectiveness of the approach. Besides, the simulation results indicate that the proposed design techniques can stabilize the quadrotor helicopter with better performance than established linear design techniques.
A simple extension of contraction theory to study incremental stability properties
DEFF Research Database (Denmark)
Jouffroy, Jerome
Contraction theory is a recent tool enabling to study the stability of nonlinear systems trajectories with respect to one another, and therefore belongs to the class of incremental stability methods. In this paper, we extend the original definition of contraction theory to incorporate...... in an explicit manner the control input of the considered system. Such an extension, called universal contraction, is quite analogous in spirit to the well-known Input-to-State Stability (ISS). It serves as a simple formulation of incremental ISS, external stability, and detectability in a differential setting....... The hierarchical combination result of contraction theory is restated in this framework, and a differential small-gain theorem is derived from results already available in Lyapunov theory....
Directory of Open Access Journals (Sweden)
M. Hassan Tanveer
2013-01-01
Full Text Available This article presents a control approach to obtain the better stabilization in attitude and altitude of quad-rotor under different disturbance conditions. In the standard Quad-rotor rotor type UAV, controlling of attitude and altitude is one of the most critical tasks and appropriate controller for stabilization of UAV is essential and necessary. These two controls under various conditions of disturbances was a field of research stimulating for the researchers. The controller proposed is contingent on the PID feedback structure with Extended Kalman Filter (EKF. From Lyapunov Stability Theorem, it is proved that quad-rotor proposed altitude control system is asymptotic as well exponentially stability. Extended Kalman Filter (EKF is used to filter out the sensors and system noises. Finally, the simulations carried out on MATLAB and the result proved the effectiveness of proposed recommended method for stabilization of attitude and altitude of quad-rotor.
Adaptive Fuzzy-Lyapunov Controller Using Biologically Inspired Swarm Intelligence
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Alejandro Carrasco Elizalde
2008-01-01
Full Text Available The collective behaviour of swarms produces smarter actions than those achieved by a single individual. Colonies of ants, flocks of birds and fish schools are examples of swarms interacting with their environment to achieve a common goal. This cooperative biological intelligence is the inspiration for an adaptive fuzzy controller developed in this paper. Swarm intelligence is used to adjust the parameters of the membership functions used in the adaptive fuzzy controller. The rules of the controller are designed using a computing-with-words approach called Fuzzy-Lyapunov synthesis to improve the stability and robustness of an adaptive fuzzy controller. Computing-with-words provides a powerful tool to manipulate numbers and symbols, like words in a natural language.
A Lyapunov theory based UPFC controller for power flow control
Energy Technology Data Exchange (ETDEWEB)
Zangeneh, Ali; Kazemi, Ahad; Hajatipour, Majid; Jadid, Shahram [Center of Excellence for Power Systems Automation and Operation, Iran University of Science and Technology, Tehran (Iran)
2009-09-15
Unified power flow controller (UPFC) is the most comprehensive multivariable device among the FACTS controllers. Capability of power flow control is the most important responsibility of UPFC. According to high importance of power flow control in transmission lines, the proper controller should be robust against uncertainty and disturbance and also have suitable settling time. For this purpose, a new controller is designed based on the Lyapunov theory and its stability is also evaluated. The Main goal of this paper is to design a controller which enables a power system to track reference signals precisely and to be robust in the presence of uncertainty of system parameters and disturbances. The performance of the proposed controller is simulated on a two bus test system and compared with a conventional PI controller. The simulation results show the power and accuracy of the proposed controller. (author)
Flexible Lyapunov Functions and Applications to Fast Mechatronic Systems
Directory of Open Access Journals (Sweden)
Mircea Lazar
2010-03-01
Full Text Available The property that every control system should posses is stability, which translates into safety in real-life applications. A central tool in systems theory for synthesizing control laws that achieve stability are control Lyapunov functions (CLFs. Classically, a CLF enforces that the resulting closed-loop state trajectory is contained within a cone with a fixed, predefined shape, and which is centered at and converges to a desired converging point. However, such a requirement often proves to be overconservative, which is why most of the real-time controllers do not have a stability guarantee. Recently, a novel idea that improves the design of CLFs in terms of flexibility was proposed. The focus of this new approach is on the design of optimization problems that allow certain parameters that define a cone associated with a standard CLF to be decision variables. In this way non-monotonicity of the CLF is explicitly linked with a decision variable that can be optimized on-line. Conservativeness is significantly reduced compared to classical CLFs, which makes flexible CLFs more suitable for stabilization of constrained discrete-time nonlinear systems and real-time control. The purpose of this overview is to highlight the potential of flexible CLFs for real-time control of fast mechatronic systems, with sampling periods below one millisecond, which are widely employed in aerospace and automotive applications.
Wang, Qiqi; Rigas, Georgios; Esclapez, Lucas; Magri, Luca; Blonigan, Patrick
2016-11-01
Bluff body flows are of fundamental importance to many engineering applications involving massive flow separation and in particular the transport industry. Coherent flow structures emanating in the wake of three-dimensional bluff bodies, such as cars, trucks and lorries, are directly linked to increased aerodynamic drag, noise and structural fatigue. For low Reynolds laminar and transitional regimes, hydrodynamic stability theory has aided the understanding and prediction of the unstable dynamics. In the same framework, sensitivity analysis provides the means for efficient and optimal control, provided the unstable modes can be accurately predicted. However, these methodologies are limited to laminar regimes where only a few unstable modes manifest. Here we extend the stability analysis to low-dimensional chaotic regimes by computing the Lyapunov covariant vectors and their associated Lyapunov exponents. We compare them to eigenvectors and eigenvalues computed in traditional hydrodynamic stability analysis. Computing Lyapunov covariant vectors and Lyapunov exponents also enables the extension of sensitivity analysis to chaotic flows via the shadowing method. We compare the computed shadowing sensitivities to traditional sensitivity analysis. These Lyapunov based methodologies do not rely on mean flow assumptions, and are mathematically rigorous for calculating sensitivities of fully unsteady flow simulations.
Analysis of stability boundaries of satellite's equilibrium attitude in a circular orbit
Novikov, M. A.
2016-03-01
An asymmetric satellite equipped with control momentum gyroscopes (CMGs) with the center of mass of the system moving uniformly in a circular orbit was considered. The stability of a relative equilibrium attitude of the satellite was analyzed using Lyapunov's direct method. The Lyapunov function V is a positive definite integral of the total energy of the perturbed motion of the system. The asymptotic stability analysis of the stationary motion of the conservative system was based on the Barbashin-Krasovskii theorem on the nonexistence of integer trajectories of the set dot V, which was obtained using the differential equations of motion of the satellite with CMGs. By analyzing the sign definiteness of the quadratic part of V, it was found earlier by V.V. Sazonov that the stability region is described by four strict inequalities. The asymptotic stability at the stability boundary was analyzed by sequentially turning these inequalities into equalities with terms of orders higher than the second taken into account in V. The sign definiteness analysis of the inhomogeneous function V at the stability boundary involved a huge amount of computations related to the multiplication, expansion, substitution, and factorization of symbolic expressions. The computations were performed by applying a computer algebra system on a personal computer.
ON RANGE DECOMPOSITION THEOREMS
Institute of Scientific and Technical Information of China (English)
吴利生
1990-01-01
We prove the following theorems.Theorem 1.Suppose f:X→Y is a closed map.X is a ωγ and β space,then Y=Y0∪(∪n=1∞Yn),where f-1(y) is countably compact for each y ∈Y0 and Yn is closed discrete in Y for each n≥1,Theorem 2-3:Suppose f:X→Y is a closed map,X is stratifable space,then Y=Y0 ∪(∪n=1∞Yn),where f-1(y) is compact for each y∈Y0 and Yn is closed discrete in Y for each n≥1.
D'Agostini, G
2005-01-01
It is curious to learn that Enrico Fermi knew how to base probabilistic inference on Bayes theorem, and that some influential notes on statistics for physicists stem from what the author calls elsewhere, but never in these notes, {\\it the Bayes Theorem of Fermi}. The fact is curious because the large majority of living physicists, educated in the second half of last century -- a kind of middle age in the statistical reasoning -- never heard of Bayes theorem during their studies, though they have been constantly using an intuitive reasoning quite Bayesian in spirit. This paper is based on recollections and notes by Jay Orear and on Gauss' ``Theoria motus corporum coelestium'', being the {\\it Princeps mathematicorum} remembered by Orear as source of Fermi's Bayesian reasoning.
Converse Barrier Certificate Theorem
DEFF Research Database (Denmark)
Wisniewski, Rafael; Sloth, Christoffer
2013-01-01
This paper presents a converse barrier certificate theorem for a generic dynamical system.We show that a barrier certificate exists for any safe dynamical system defined on a compact manifold. Other authors have developed a related result, by assuming that the dynamical system has no singular...... points in the considered subset of the state space. In this paper, we redefine the standard notion of safety to comply with generic dynamical systems with multiple singularities. Afterwards, we prove the converse barrier certificate theorem and illustrate the differences between ours and previous work...
Kartavtsev, Alexander
2014-01-01
According to the Goldstone theorem a scalar theory with a spontaneously broken global symmetry contains strictly massless states. In this letter we identify a loophole in the current-algebra proof of the theorem. Therefore, the question whether in models with Mexican hat potential the tangential excitations are strictly massless or are just almost massless as compared to the radial ones remains open. We also argue that mass of the tangential excitations approaches zero even if the symmetry is not spontaneously broken but a combination of the field components invariant under the symmetry transformations acquires a large vacuum expectation value.
Pérez-Espigares, Carlos; Redig, Frank; Giardinà, Cristian
2015-08-01
For non-equilibrium systems of interacting particles and for interacting diffusions in d-dimensions, a novel fluctuation relation is derived. The theorem establishes a quantitative relation between the probabilities of observing two current values in different spatial directions. The result is a consequence of spatial symmetries of the microscopic dynamics, generalizing in this way the Gallavotti-Cohen fluctuation theorem related to the time-reversal symmetry. This new perspective opens up the possibility of direct experimental measurements of fluctuation relations of vectorial observables.
Hydrodynamische Lyapunov-Moden in mehrkomponentigen Lennard-Jones-Flüssigkeiten
Drobniewski, Christian
2011-01-01
Die Charakterisierung hochdimensionaler Systeme mit Lyapunov-Instabilität wird durch das Lyapunov-Spektrum und die zugehörigen Lyapunov-Vektoren ermöglicht. Für eine Vielzahl von derartigen Systemen (Coupled-Map-Lattices, Hartkugel-Systeme, Systeme mit ausgedehnten Potentialen ...) konnte durch die Untersuchung der Lyapunov-Vektoren die Existenz von hydrodynamischen Lyapunov-Moden nachgewiesen werden. Diese kollektiven Anregungen zeigen sich in Lyapunov-Vektoren, deren Lyapunov-Exponenten de...
Ferromagnetism beyond Lieb's theorem
Costa, Natanael C.; Mendes-Santos, Tiago; Paiva, Thereza; Santos, Raimundo R. dos; Scalettar, Richard T.
2016-10-01
The noninteracting electronic structures of tight-binding models on bipartite lattices with unequal numbers of sites in the two sublattices have a number of unique features, including the presence of spatially localized eigenstates and flat bands. When a uniform on-site Hubbard interaction U is turned on, Lieb proved rigorously that at half-filling (ρ =1 ) the ground state has a nonzero spin. In this paper we consider a "CuO2 lattice" (also known as "Lieb lattice," or as a decorated square lattice), in which "d orbitals" occupy the vertices of the squares, while "p orbitals" lie halfway between two d orbitals; both d and p orbitals can accommodate only up to two electrons. We use exact determinant quantum Monte Carlo (DQMC) simulations to quantify the nature of magnetic order through the behavior of correlation functions and sublattice magnetizations in the different orbitals as a function of U and temperature; we have also calculated the projected density of states, and the compressibility. We study both the homogeneous (H) case, Ud=Up , originally considered by Lieb, and the inhomogeneous (IH) case, Ud≠Up . For the H case at half-filling, we found that the global magnetization rises sharply at weak coupling, and then stabilizes towards the strong-coupling (Heisenberg) value, as a result of the interplay between the ferromagnetism of like sites and the antiferromagnetism between unlike sites; we verified that the system is an insulator for all U . For the IH system at half-filling, we argue that the case Up≠Ud falls under Lieb's theorem, provided they are positive definite, so we used DQMC to probe the cases Up=0 ,Ud=U and Up=U ,Ud=0 . We found that the different environments of d and p sites lead to a ferromagnetic insulator when Ud=0 ; by contrast, Up=0 leads to to a metal without any magnetic ordering. In addition, we have also established that at density ρ =1 /3 , strong antiferromagnetic correlations set in, caused by the presence of one fermion on each
Nonuniform exponential dichotomies and Lyapunov functions
Barreira, Luis; Dragičević, Davor; Valls, Claudia
2017-05-01
For the nonautonomous dynamics defined by a sequence of bounded linear operators acting on an arbitrary Hilbert space, we obtain a characterization of the notion of a nonuniform exponential dichotomy in terms of quadratic Lyapunov sequences. We emphasize that, in sharp contrast with previous results, we consider the general case of possibly noninvertible linear operators, thus requiring only the invertibility along the unstable direction. As an application, we give a simple proof of the robustness of a nonuniform exponential dichotomy under sufficiently small linear perturbations.
Lyapunov exponents for infinite dimensional dynamical systems
Mhuiris, Nessan Mac Giolla
1987-01-01
Classically it was held that solutions to deterministic partial differential equations (i.e., ones with smooth coefficients and boundary data) could become random only through one mechanism, namely by the activation of more and more of the infinite number of degrees of freedom that are available to such a system. It is only recently that researchers have come to suspect that many infinite dimensional nonlinear systems may in fact possess finite dimensional chaotic attractors. Lyapunov exponents provide a tool for probing the nature of these attractors. This paper examines how these exponents might be measured for infinite dimensional systems.
Diverging Fluctuations of the Lyapunov Exponents.
Pazó, Diego; López, Juan M; Politi, Antonio
2016-07-15
We show that in generic one-dimensional Hamiltonian lattices the diffusion coefficient of the maximum Lyapunov exponent diverges in the thermodynamic limit. We trace this back to the long-range correlations associated with the evolution of the hydrodynamic modes. In the case of normal heat transport, the divergence is even stronger, leading to the breakdown of the usual single-function Family-Vicsek scaling ansatz. A similar scenario is expected to arise in the evolution of rough interfaces in the presence of suitably correlated background noise.
On fuzzy sampled-data control of chaotic systems via a time-dependent Lyapunov functional approach.
Wang, Zi-Peng; Wu, Huai-Ning
2015-04-01
In this paper, a novel approach to fuzzy sampled-data control of chaotic systems is presented by using a time-dependent Lyapunov functional. The advantage of the new method is that the Lyapunov functional is continuous at sampling times but not necessarily positive definite inside the sampling intervals. Compared with the existing works, the constructed Lyapunov functional makes full use of the information on the piecewise constant input and the actual sampling pattern. In terms of a new parameterized linear matrix inequality (LMI) technique, a less conservative stabilization condition is derived to guarantee the exponential stability for the closed-loop fuzzy sampled-data system. By solving a set of LMIs, the fuzzy sampled-data controller can be easily obtained. Finally, the chaotic Lorenz system and Rössler's system are employed to illustrate the feasibility and effectiveness of the proposed method.
Virial Theorem and Scale Transformations.
Kleban, Peter
1979-01-01
Discussed is the virial theorem, which is useful in classical, quantum, and statistical mechanics. Two types of derivations of this theorem are presented and the relationship between the two is explored. (BT)
A New Approach to the Method of Lyapunov Functionals and Its Applications
Directory of Open Access Journals (Sweden)
Yunguo Jin
2013-01-01
Full Text Available We show some results which can replace the graph theory used to construct global Lyapunov functions in some coupled systems of differential equations. We present an example of an epidemic model with stage structure and latency spreading in a heterogeneous host population and obtain a more general threshold for the extinction and persistence of a disease. Using some results obtained by mathematical induction and suitable Lyapunov functionals, we prove the global stability of the endemic equilibrium. For some coupled systems of differential equations, by a similar approach to the discussion of the epidemic model, the conditions of threshold property or global stability can be established without the assumption that the relative matrix is irreducible.
A unified perspective on robot control - The energy Lyapunov function approach
Wen, John T.
1990-01-01
A unified framework for the stability analysis of robot tracking control is presented. By using an energy-motivated Lyapunov function candidate, the closed-loop stability is shown for a large family of control laws sharing a common structure of proportional and derivative feedback and a model-based feedforward. The feedforward can be zero, partial or complete linearized dynamics, partial or complete nonlinear dynamics, or linearized or nonlinear dynamics with parameter adaptation. As result, the dichotomous approaches to the robot control problem based on the open-loop linearization and nonlinear Lyapunov analysis are both included in this treatment. Furthermore, quantitative estimates of the trade-offs between different schemes in terms of the tracking performance, steady state error, domain of convergence, realtime computation load and required a prior model information are derived.
Flexible Lyapunov Functions and Applications to Fast Mechatronic Systems
Lazar, M
2010-01-01
The property that every control system should posses is stability, which translates into safety in real-life applications. A central tool in systems theory for synthesizing control laws that achieve stability are control Lyapunov functions (CLFs). Classically, a CLF enforces that the resulting closed-loop state trajectory is contained within a cone with a fixed, predefined shape, and which is centered at and converges to a desired converging point. However, such a requirement often proves to be overconservative, which is why most of the real-time controllers do not have a stability guarantee. Recently, a novel idea that improves the design of CLFs in terms of flexibility was proposed. The focus of this new approach is on the design of optimization problems that allow certain parameters that define a cone associated with a standard CLF to be decision variables. In this way non-monotonicity of the CLF is explicitly linked with a decision variable that can be optimized on-line. Conservativeness is significantly ...
Certified Kruskal's Tree Theorem
Directory of Open Access Journals (Sweden)
Christian Sternagel
2014-07-01
Full Text Available This article presents the first formalization of Kurskal's tree theorem in aproof assistant. The Isabelle/HOL development is along the lines of Nash-Williams' original minimal bad sequence argument for proving the treetheorem. Along the way, proofs of Dickson's lemma and Higman's lemma, as well as some technical details of the formalization are discussed.
DEFF Research Database (Denmark)
Thomassen, Carsten
2004-01-01
We present a short proof of the theorem of Tutte that every planar 3-connected graph has a drawing in the plane such that every vertex which is not on the outer cycle is the barycenter of its neighbors. Moreover, this holds for any prescribed representation of the outer cycle. (C) 2004 Wiley Peri...
DEFF Research Database (Denmark)
Törnquist, Asger Dag; Weiss, W.
2009-01-01
We prove the following descriptive set-theoretic analogue of a theorem of R. 0. Davies: Every σ function f:ℝ × ℝ → ℝ can be represented as a sum of rectangular Σ functions if and only if all reals are constructible. © Instytut Matematyczny PAN, 2009....
Rediscovering Schreinemakers' Theorem.
Bathurst, Bruce
1983-01-01
Schreinemakers' theorem (arrangement of curves around an invariant point), derived from La Chatelier's principle, can be rediscovered by students asked to use the principle when solving a natural problem such as "How does diluting a mineral/fluid alter shape of a pressure/temperature diagram?" Background information and instructional…
Teleman, Nicolae
2011-01-01
$Local^{3}$ Index Theorem means $Local(Local(Local \\;Index \\; Theorem)))$. $Local \\; Index \\; Theorem$ is the Connes-Moscovici local index theorem \\cite{Connes-Moscovici1}, \\cite{Connes-Moscovici2}. The second "Local" refers to the cyclic homology localised to a certain separable subring of the ground algebra, while the last one refers to Alexander-Spanier type cyclic homology. The Connes-Moscovici work is based on the operator $R(A) = \\mathbf{P} - \\mathbf{e}$ associated to the elliptic pseudo-differential operator $A$ on the smooth manifold $M$, where $\\mathbf{P}$, $\\mathbf{e}$ are idempotents, see \\cite{Connes-Moscovici1}, Pg. 353. The operator $R(A)$ has two main merits: it is a smoothing operator and its distributional kernel is situated in an arbitrarily small neighbourhood of the diagonal in $M \\times M$. The operator $R(A)$ has also two setbacks: -i) it is not an idempotent (and therefore it does not have a genuine Connes-Chern character); -ii) even if it were an idempotent, its Connes-Chern character ...
Multivariate irregular sampling theorem
Institute of Scientific and Technical Information of China (English)
无
2009-01-01
In this paper,we prove a Marcinkiewicz-Zygmund type inequality for multivariate entire functions of exponential type with non-equidistant spaced sampling points. And from this result,we establish a multivariate irregular Whittaker-Kotelnikov-Shannon type sampling theorem.
Multivariate irregular sampling theorem
Institute of Scientific and Technical Information of China (English)
CHEN GuangGui; FANG GenSun
2009-01-01
In this paper, we prove a Marcinkiewicz-Zygmund type inequality for multivariate entire functions of exponential type with non-equidistant spaced sampling points. And from this result, we establish a multivariate irregular Whittaker-Kotelnikov-Shannon type sampling theorem.
Dalen, D. van
2008-01-01
The following pages make form a new chapter for the book Logic and Structure. This chapter deals with the incompleteness theorem, and contains enough basic material for the treatment of the required notions of computability, representability and the like. This chapter will appear in the next edition
Converse Barrier Certificate Theorems
DEFF Research Database (Denmark)
Wisniewski, Rafael; Sloth, Christoffer
2016-01-01
This paper shows that a barrier certificate exists for any safe dynamical system. Specifically, we prove converse barrier certificate theorems for a class of structurally stable dynamical systems. Other authors have developed a related result by assuming that the dynamical system has neither sing...
Automated Discovery of Inductive Theorems
McCasland, Roy; Bundy, Alan; Serge, Autexier
2007-01-01
Inductive mathematical theorems have, as a rule, historically been quite difficult to prove – both for mathematics students and for auto- mated theorem provers. That said, there has been considerable progress over the past several years, within the automated reasoning community, towards proving some of these theorems. However, little work has been done thus far towards automatically discovering them. In this paper we present our methods of discovering (as well as proving) inductive theorems, ...
Statistics of Lyapunov exponent spectrum in randomly coupled Kuramoto oscillators.
Patra, Soumen K; Ghosh, Anandamohan
2016-03-01
Characterization of spatiotemporal dynamics of coupled oscillatory systems can be done by computing the Lyapunov exponents. We study the spatiotemporal dynamics of randomly coupled network of Kuramoto oscillators and find that the spectral statistics obtained from the Lyapunov exponent spectrum show interesting sensitivity to the coupling matrix. Our results indicate that in the weak coupling limit the gap distribution of the Lyapunov spectrum is Poissonian, while in the limit of strong coupling the gap distribution shows level repulsion. Moreover, the oscillators settle to an inhomogeneous oscillatory state, and it is also possible to infer the random network properties from the Lyapunov exponent spectrum.
The Second Noether Theorem on Time Scales
Malinowska, Agnieszka B.; Natália Martins
2013-01-01
We extend the second Noether theorem to variational problems on time scales. As corollaries we obtain the classical second Noether theorem, the second Noether theorem for the $h$ -calculus and the second Noether theorem for the $q$ -calculus.
An Extension of Sobolev's Theorem
Institute of Scientific and Technical Information of China (English)
无
2000-01-01
Sobolev's Theorem is the most fundamental theorem in the theory of Invariant Cubature Formulas (ICFs). In this paper, a quantitative structure is established for the classical ICFs. Enlightened by this structure, the author generalizes the concept of ICFs and extends the Sobolev's Theorem to the case of generalized ICFs. Several illustrative examples are given.
Russell, Alan R.
2004-01-01
Pick's theorem can be used in various ways just like a lemon. This theorem generally finds its way in the syllabus approximately at the middle school level and in fact at times students have even calculated the area of a state considering its outline with the help of the above theorem.
Generalized no-broadcasting theorem.
Barnum, Howard; Barrett, Jonathan; Leifer, Matthew; Wilce, Alexander
2007-12-14
We prove a generalized version of the no-broadcasting theorem, applicable to essentially any nonclassical finite-dimensional probabilistic model satisfying a no-signaling criterion, including ones with "superquantum" correlations. A strengthened version of the quantum no-broadcasting theorem follows, and its proof is significantly simpler than existing proofs of the no-broadcasting theorem.
Virial theorem and hypervirial theorem in a spherical geometry
Energy Technology Data Exchange (ETDEWEB)
Li Yan; Chen Jingling [Theoretical Physics Division, Chern Institute of Mathematics, Nankai University, Tianjin 300071 (China); Zhang Fulin, E-mail: flzhang@tju.edu.cn, E-mail: chenjl@nankai.edu.cn [Physics Department, School of Science, Tianjin University, Tianjin 300072 (China)
2011-09-09
The virial theorem in the one- and two-dimensional spherical geometry are presented in both classical and quantum mechanics. Choosing a special class of hypervirial operators, the quantum hypervirial relations in the spherical spaces are obtained. With the aid of the Hellmann-Feynman theorem, these relations can be used to formulate a perturbation theorem without wavefunctions, corresponding to the hypervirial-Hellmann-Feynman theorem perturbation theorem of Euclidean geometry. The one-dimensional harmonic oscillator and two-dimensional Coulomb system in the spherical spaces are given as two sample examples to illustrate the perturbation method. (paper)
Using Lyapunov exponents to predict the onset of chaos in nonlinear oscillators.
Ryabov, Vladimir B
2002-07-01
An analytic technique for predicting the emergence of chaotic instability in nonlinear nonautonomous dissipative oscillators is proposed. The method is based on the Lyapunov-type stability analysis of an arbitrary phase trajectory and the standard procedure of calculating the Lyapunov characteristic exponents. The concept of temporally local Lyapunov exponents is then utilized for specifying the area in the phase space where any trajectory is asymptotically stable, and, therefore, the existence of chaotic attractors is impossible. The procedure of linear coordinate transform optimizing the linear part of the vector field is developed for the purpose of maximizing the stability area in the vicinity of a stable fixed point. By considering the inverse conditions of asymptotic stability, this approach allows formulating a necessary condition for chaotic motion in a broad class of nonlinear oscillatory systems, including many cases of practical interest. The examples of externally excited one- and two-well Duffing oscillators and a planar pendulum demonstrate efficiency of the proposed method, as well as a good agreement of the theoretical predictions with the results of numerical experiments. The comparison of the proposed method with Melnikov's criterion shows a potential advantage of using the former one at high values of dissipation parameter and/or multifrequency type of excitation in dynamical systems.
A reconstruction theorem for almost-commutative spectral triples
Ćaćić, Branimir
2011-01-01
We propose an expansion of the definition of almost-commutative spectral triple that accommodates non-trivial fibrations and is stable under inner fluctuation of the metric, and then prove a reconstruction theorem for almost-commutative spectral triples under this definition as a simple consequence of the reconstruction theorem for commutative spectral triples. Along the way, we weaken the orientability hypothesis in Connes's reconstruction theorem for commutative spectral triples, and, following Chakraborty and Mathai, prove a number of results concerning the stability of properties of spectral triples under suitable perturbation of the Dirac operator.
Stability Analysis of Grasps with a Robotic Multifingered Hand
Institute of Scientific and Technical Information of China (English)
WAN An-hua
2005-01-01
Stability is a significant property for a robot hand grasp to perform complex tasks similar to human hands. The common method to investigate the stability of robotic multi-fingered grasp system is Lyapunov direct method, but usually it is rather difficult to construct a proper Lyapunov function. Avoiding the hard work of constructing a Lyapunov function, we propose the sufficient conditions for stability of the robotic grasp system.
Taylor, Marika
2016-01-01
The F theorem states that, for a unitary three dimensional quantum field theory, the F quantity defined in terms of the partition function on a three sphere is positive, stationary at fixed point and decreases monotonically along a renormalization group flow. We construct holographic renormalization group flows corresponding to relevant deformations of three-dimensional conformal field theories on spheres, working to quadratic order in the source. For these renormalization group flows, the F quantity at the IR fixed point is always less than F at the UV fixed point, but F increases along the RG flow for deformations by operators of dimension $3/2 < \\Delta < 5/2$. Therefore the strongest version of the F theorem is in general violated.
Indian Academy of Sciences (India)
N V Rao
2003-02-01
The general theme of this note is illustrated by the following theorem: Theorem 1. Suppose is a compact set in the complex plane and 0 belongs to the boundary . Let $\\mathcal{A}(K)$ denote the space of all functions on such that is holomorphic in a neighborhood of and (0) = 0. Also for any given positive integer , let $\\mathcal{A}(m, K)$ denote the space of all such that is holomorphic in a neighborhood of and $f(0) = f'(0) = \\cdots = f^{(m)}(0) = 0$. Then $\\mathcal{A}(m, K)$ is dense in $\\mathcal{A}(K)$ under the supremum norm on provided that there exists a sector $W = \\{re^{i}; 0 ≤ r ≤ , ≤ ≤ \\}$ such that $W \\cap K = \\{0\\}$. (This is the well-known Poincare's external cone condition).} We present various generalizations of this result in the context of higher dimensions replacing holomorphic with harmonic.
The Lyapunov exponents of the Van der Pol oscillator
Grasman, J.; Verhulst, F.; Shih, S.D.
2005-01-01
Lyapunov exponents characterize the dynamics of a system near its attractor. For the Van der Pol oscillator these are quantities for which an approximation should be at hand. Similar to the asymptotic approximation of amplitude and period, expressions are derived for the non-zero Lyapunov exponent
Calculating Lyapunov Exponents: Applying Products and Evaluating Integrals
McCartney, Mark
2010-01-01
Two common examples of one-dimensional maps (the tent map and the logistic map) are generalized to cases where they have more than one control parameter. In the case of the tent map, this still allows the global Lyapunov exponent to be found analytically, and permits various properties of the resulting global Lyapunov exponents to be investigated…
Calculating Lyapunov Exponents: Applying Products and Evaluating Integrals
McCartney, Mark
2010-01-01
Two common examples of one-dimensional maps (the tent map and the logistic map) are generalized to cases where they have more than one control parameter. In the case of the tent map, this still allows the global Lyapunov exponent to be found analytically, and permits various properties of the resulting global Lyapunov exponents to be investigated…
1987-03-20
with standard expressions of spherical trigonometry is sinr)0 = cos0 sini//0 (4.37) which is consistent with the results obtained previously with...theorems for discrete transforms. However, sampling questions inlroduce difficult obstacles in the develop- ment of a discrete theory. First, sampling...additional obstacle to discrete represen- tations of the CT. An example of qualitative predication of the shape of silhouettes with the Silhouette-Slice
Gouin, Henri
2015-01-01
Comments on Archimedes' theorem about sphere and cylinder; In his treatise addressed to Dositheus of Pelusium, Archimedes of Syracuse obtained the result of which he was the most proud: a sphere has two-thirds the volume of its circumscribing cylinder. At his request a sculpted sphere and cylinder were placed on his tomb near Syracuse. Usually, it is admitted that to find this formula, Archimedes used a half polygon inscribed in a semicircle; then he performed rotations of these two figures t...
Sandwich classification theorem
Directory of Open Access Journals (Sweden)
Alexey Stepanov
2015-09-01
Full Text Available The present note arises from the author's talk at the conference ``Ischia Group Theory 2014''. For subgroups FleN of a group G denote by Lat(F,N the set of all subgroups of N , containing F . Let D be a subgroup of G . In this note we study the lattice LL=Lat(D,G and the lattice LL ′ of subgroups of G , normalized by D . We say that LL satisfies sandwich classification theorem if LL splits into a disjoint union of sandwiches Lat(F,N G (F over all subgroups F such that the normal closure of D in F coincides with F . Here N G (F denotes the normalizer of F in G . A similar notion of sandwich classification is introduced for the lattice LL ′ . If D is perfect, i.,e. coincides with its commutator subgroup, then it turns out that sandwich classification theorem for LL and LL ′ are equivalent. We also show how to find basic subroup F of sandwiches for LL ′ and review sandwich classification theorems in algebraic groups over rings.
Preparing entangled states by Lyapunov control
Shi, Z. C.; Wang, L. C.; Yi, X. X.
2016-09-01
By Lyapunov control, we present a protocol to prepare entangled states such as Bell states in the context of cavity QED system. The advantage of our method is of threefold. Firstly, we can only control the phase of classical fields to complete the preparation process. Secondly, the evolution time is sharply shortened when compared to adiabatic control. Thirdly, the final state is steady after removing control fields. The influence of decoherence caused by the atomic spontaneous emission and the cavity decay is discussed. The numerical results show that the control scheme is immune to decoherence, especially for the atomic spontaneous emission from |2rangle to |1rangle . This can be understood as the state staying in an invariant subspace. Finally, we generalize this method in preparation of W state.
Preparation of topological modes by Lyapunov control.
Shi, Z C; Zhao, X L; Yi, X X
2015-09-08
By Lyapunov control, we present a proposal to drive quasi-particles into a topological mode in quantum systems described by a quadratic Hamiltonian. The merit of this control is the individual manipulations on the boundary sites. We take the Kitaev's chain as an illustration for Fermi systems and show that an arbitrary excitation mode can be steered into the Majorana zero mode by manipulating the chemical potential of the boundary sites. For Bose systems, taking the noninteracting Su-Schrieffer-Heeger (SSH) model as an example, we illustrate how to drive the system into the edge mode. The sensitivity of the fidelity to perturbations and uncertainties in the control fields and initial modes is also examined. The experimental feasibility of the proposal and the possibility to replace the continuous control field with square wave pulses is finally discussed.
Preparing entangled states by Lyapunov control
Shi, Z. C.; Wang, L. C.; Yi, X. X.
2016-12-01
By Lyapunov control, we present a protocol to prepare entangled states such as Bell states in the context of cavity QED system. The advantage of our method is of threefold. Firstly, we can only control the phase of classical fields to complete the preparation process. Secondly, the evolution time is sharply shortened when compared to adiabatic control. Thirdly, the final state is steady after removing control fields. The influence of decoherence caused by the atomic spontaneous emission and the cavity decay is discussed. The numerical results show that the control scheme is immune to decoherence, especially for the atomic spontaneous emission from |2rangle to |1rangle . This can be understood as the state staying in an invariant subspace. Finally, we generalize this method in preparation of W state.
Numerical solution of large Lyapunov equations
Saad, Youcef
1989-01-01
A few methods are proposed for solving large Lyapunov equations that arise in control problems. The common case where the right hand side is a small rank matrix is considered. For the single input case, i.e., when the equation considered is of the form AX + XA(sup T) + bb(sup T) = 0, where b is a column vector, the existence of approximate solutions of the form X = VGV(sup T) where V is N x m and G is m x m, with m small is established. The first class of methods proposed is based on the use of numerical quadrature formulas, such as Gauss-Laguerre formulas, applied to the controllability Grammian. The second is based on a projection process of Galerkin type. Numerical experiments are presented to test the effectiveness of these methods for large problems.
Experimentally realizable control fields in quantum Lyapunov control
Yi, X X; Wu, Chunfeng; Feng, X L; Oh, C H
2011-01-01
As a hybrid of techniques from open-loop and feedback control, Lyapunov control has the advantage that it is free from the measurement-induced decoherence but it includes the system's instantaneous message in the control loop. Often, the Lyapunov control is confronted with time delay in the control fields and difficulty in practical implementations of the control. In this paper, we study the effect of time-delay on the Lyapunov control, and explore the possibility of replacing the control field with a pulse train or a bang-bang signal. The efficiency of the Lyapunov control is also presented through examining the convergence time of the controlled system. These results suggest that the Lyapunov control is robust gainst time delay, easy to realize and effective for high-dimensional quantum systems.
Generalized Lyapunov exponent as a unified characterization of dynamical instabilities.
Akimoto, Takuma; Nakagawa, Masaki; Shinkai, Soya; Aizawa, Yoji
2015-01-01
The Lyapunov exponent characterizes an exponential growth rate of the difference of nearby orbits. A positive Lyapunov exponent (exponential dynamical instability) is a manifestation of chaos. Here, we propose the Lyapunov pair, which is based on the generalized Lyapunov exponent, as a unified characterization of nonexponential and exponential dynamical instabilities in one-dimensional maps. Chaos is classified into three different types, i.e., superexponential, exponential, and subexponential chaos. Using one-dimensional maps, we demonstrate superexponential and subexponential chaos and quantify the dynamical instabilities by the Lyapunov pair. In subexponential chaos, we show superweak chaos, which means that the growth of the difference of nearby orbits is slower than a stretched exponential growth. The scaling of the growth is analytically studied by a recently developed theory of a continuous accumulation process, which is related to infinite ergodic theory.
Lyapunov exponent diagrams of a 4-dimensional Chua system.
Stegemann, Cristiane; Albuquerque, Holokx A; Rubinger, Rero M; Rech, Paulo C
2011-09-01
We report numerical results on the existence of periodic structures embedded in chaotic and hyperchaotic regions on the Lyapunov exponent diagrams of a 4-dimensional Chua system. The model was obtained from the 3-dimensional Chua system by the introduction of a feedback controller. Both the largest and the second largest Lyapunov exponents were considered in our colorful Lyapunov exponent diagrams, and allowed us to characterize periodic structures and regions of chaos and hyperchaos. The shrimp-shaped periodic structures appear to be malformed on some of Lyapunov exponent diagrams, and they present two different bifurcation scenarios to chaos when passing the boundaries of itself, namely via period-doubling and crisis. Hyperchaos-chaos transition can also be observed on the Lyapunov exponent diagrams for the second largest exponent.
Directory of Open Access Journals (Sweden)
Ruofeng Rao
2013-01-01
Full Text Available The robust exponential stability of delayed fuzzy Markovian-jumping Cohen-Grossberg neural networks (CGNNs with nonlinear p-Laplace diffusion is studied. Fuzzy mathematical model brings a great difficulty in setting up LMI criteria for the stability, and stochastic functional differential equations model with nonlinear diffusion makes it harder. To study the stability of fuzzy CGNNs with diffusion, we have to construct a Lyapunov-Krasovskii functional in non-matrix form. But stochastic mathematical formulae are always described in matrix forms. By way of some variational methods in W1,p(Ω, Itô formula, Dynkin formula, the semi-martingale convergence theorem, Schur Complement Theorem, and LMI technique, the LMI-based criteria on the robust exponential stability and almost sure exponential robust stability are finally obtained, the feasibility of which can efficiently be computed and confirmed by computer MatLab LMI toolbox. It is worth mentioning that even corollaries of the main results of this paper improve some recent related existing results. Moreover, some numerical examples are presented to illustrate the effectiveness and less conservatism of the proposed method due to the significant improvement in the allowable upper bounds of time delays.
Direct Lyapunov-based control law design for spacecraft attitude maneuvers
Institute of Scientific and Technical Information of China (English)
HU Likun; ANG Qingchao
2006-01-01
A direct Lyapunov-based control law is presented to perform on-orbit stability for spacecraft attitude maneuvers. Spacecraft attitude kinematic equations and dynamic equations are coupled, nonlinear, multi-input multi-output(MIMO), which baffles controller design. Orbit angular rates are taken into account in kinematic equations and influence of gravity gradient moments and disturbance moments on the spacecraft attitude in dynamic equations is considered to approach the practical environment, which enhance the problem complexity to some extent. Based on attitude tracking errors and angular rates, a Lyapunov function is constructed, through which the stabilizing feedback control law is deduced via Lie derivation of the Lyapunov function. The proposed method can deal with the case that the spacecraft is subjected to mass property variations or centroidal inertia matrix variations due to fuel assumption or flexibility, and disturbance moments, which shows the proposed controller is robust for spacecraft attitude maneuvers. The unlimited controller and the limited controller are taken into account respectively in simulations. Simulation results are demonstrated to validate effectiveness and feasibility of the proposed method.
Singlet and triplet instability theorems
Yamada, Tomonori; Hirata, So
2015-09-01
A useful definition of orbital degeneracy—form-degeneracy—is introduced, which is distinct from the usual energy-degeneracy: Two canonical spatial orbitals are form-degenerate when the energy expectation value in the restricted Hartree-Fock (RHF) wave function is unaltered upon a two-electron excitation from one of these orbitals to the other. Form-degenerate orbitals tend to have isomorphic electron densities and occur in the highest-occupied and lowest-unoccupied molecular orbitals (HOMOs and LUMOs) of strongly correlated systems. Here, we present a mathematical proof of the existence of a triplet instability in a real or complex RHF wave function of a finite system in the space of real or complex unrestricted Hartree-Fock wave functions when HOMO and LUMO are energy- or form-degenerate. We also show that a singlet instability always exists in a real RHF wave function of a finite system in the space of complex RHF wave functions, when HOMO and LUMO are form-degenerate, but have nonidentical electron densities, or are energy-degenerate. These theorems provide Hartree-Fock-theory-based explanations of Hund's rule, a singlet instability in Jahn-Teller systems, biradicaloid electronic structures, and a triplet instability during some covalent bond breaking. They also suggest (but not guarantee) the spontaneous formation of a spin density wave (SDW) in a metallic solid. The stability theory underlying these theorems extended to a continuous orbital-energy spectrum proves the existence of an oscillating (nonspiral) SDW instability in one- and three-dimensional homogeneous electron gases, but only at low densities or for strong interactions.
A variational approach to Lyapunov type inequalities from ODEs to PDEs
Cañada, Antonio
2015-01-01
This book highlights the current state of Lyapunov-type inequalities through a detailed analysis. Aimed toward researchers and students working in differential equations and those interested in the applications of stability theory and resonant systems, the book begins with an overview Lyapunov’s original results and moves forward to include prevalent results obtained in the past ten years. Detailed proofs and an emphasis on basic ideas are provided for different boundary conditions for ordinary differential equations, including Neumann, Dirichlet, periodic, and antiperiodic conditions. Novel results of higher eigenvalues, systems of equations, partial differential equations as well as variational approaches are presented. To this respect, a new and unified variational point of view is introduced for the treatment of such problems and a systematic discussion of different types of boundary conditions is featured. Various problems make the study of Lyapunov-type inequalities of interest to those in pure and ...
Cao, Fangfei; Liu, Jinkun
2017-10-01
Considering full state constraints, this paper designs a boundary controller for a two-link rigid-flexible manipulator via Barrier Lyapunov Function. The dynamic model of the two-link rigid-flexible manipulator is described by coupled ordinary differential equations- partial differential equations (ODEs-PDEs). Based on the original model without neglecting the high-frequency modes, boundary controller is proposed to regulate the joint positions and eliminate the elastic vibration simultaneously. To ensure that the full state constraints which include position, speed and vibration constraints are not transgressed, a Barrier Lyapunov Function is employed in the proposed controller. The asymptotic stability of the closed-loop system is rigorously proved by the LaSalle's Invariance Principle. Simulations are given to verify the effectiveness of the proposed controller with state constraints.
On Designing Lyapunov-Krasovskii Based AQM for Routers Supporting TCP Flows
Labit, Yann; Gouaisbaut, Frédéric; 10.1109/CDC.2007.4434673
2009-01-01
For the last few years, we assist to a growing interest of designing AQM (Active Queue Management) using control theory. In this paper, we focus on the synthesis of an AQM based on the Lyapunov theory for time delay systems. With the help of a recently developed Lyapunov-Krasovskii functional and using a state space representation of a linearized fluid model of TCP, two robust AQMs stabilizing the TCP model are constructed. Notice that our results are constructive and the synthesis problem is reduced to a convex optimization scheme expressed in terms of linear matrix inequalities (LMIs). Finally, an example extracted from the literature and simulations via {\\it NS simulator} support our study.
Liu, Xiaoyang; Yu, Wenwu; Cao, Jinde; Chen, Shun
2015-08-01
This paper is concerned with the sampled-data state estimation and H(∞) filtering for a class of Markovian jump systems with the discontinuous Lyapunov approach. The system measurements are sampled and then transmitted to the estimator and filter in order to estimate the state of the jumped system under consideration. The corresponding error dynamics is represented by a system with two types of delays: one is from the system itself, and the other from the sampling period. As the delay due to sampling is discontinuous, a corresponding discontinuous Lyapunov functional is constructed, and sufficient conditions are established so as to guarantee both the asymptotic mean-square stability and the H(∞) performance for the filtering error systems. The explicit expressions of the desired estimator and filter are further provided. Finally, two simulation examples are given to illustrate the design procedures and performances of the proposed method.
Stability of uncertain systems. Ph.D. Thesis
Blankenship, G. L.
1971-01-01
The asymptotic properties of feedback systems are discussed, containing uncertain parameters and subjected to stochastic perturbations. The approach is functional analytic in flavor and thereby avoids the use of Markov techniques and auxiliary Lyapunov functionals characteristic of the existing work in this area. The results are given for the probability distributions of the accessible signals in the system and are proved using the Prohorov theory of the convergence of measures. For general nonlinear systems, a result similar to the small loop-gain theorem of deterministic stability theory is given. Boundedness is a property of the induced distributions of the signals and not the usual notion of boundedness in norm. For the special class of feedback systems formed by the cascade of a white noise, a sector nonlinearity and convolution operator conditions are given to insure the total boundedness of the overall feedback system.
Sun, Yeong-Jeu; Wu, Yu-Biaw; Wang, Ching-Cheng
2013-06-01
In this study, the concept of global exponential ε-stabilization is introduced and the robust stabilization for a class of nonlinear systems with single input is investigated. Based on Lyapunov-like Theorem with differential and integral inequalities, a feedback control is proposed to realize the global stabilization of such nonlinear systems with any pre-specified exponential convergence rate. The guaranteed exponential convergence rate can be also correctly estimated. This result can be straightforwardly applicable to some famous chaotic systems. Besides, it will be proven that a single and linear control, with lower dimensions than that of the states, can realize the global exponential stability of some famous chaotic systems. Finally, comparisons of our main results with recently published results as well as numerical examples with circuit realization are provided to show the effectiveness and superiority of the obtained results.
The direct Lyapunov method for the stabilisation of the Furuta pendulum
Aguilar-Ibañez, Carlos; Suárez-Castañón, Miguel S.; Gutiérres-Frias, Oscar O.
2010-11-01
A nonlinear controller for the stabilisation of the Furuta pendulum is presented. The control strategy is based on a partial feedback linearisation. In a first stage only the actuated coordinate of the Furuta pendulum is linearised. Then, the stabilising feedback controller is obtained by applying the Lyapunov direct method. That is, using this method we prove local asymptotic stability and demonstrate that the closed-loop system has a large region of attraction. The stability analysis is carried out by means of LaSalle's invariance principle. To assess the controller effectiveness, the results of the corresponding numerical simulations are presented.
Solution of the Lyapunov matrix equation for a system with a time-dependent stiffness matrix
DEFF Research Database (Denmark)
Pommer, Christian; Kliem, Wolfhard
2004-01-01
The stability of the linearized model of a rotor system with non-symmetric strain and axial loads is investigated. Since we are using a fixed reference system, the differential equations have the advantage to be free of Coriolis and centrifugal forces. A disadvantage is nevertheless the occurrenc...... of time-dependent periodic terms in the stiffness matrix. However, by solving the Lyapunov matrix equation we can formulate several stability conditions for the rotor system. Hereby the positive definiteness of a certain averaged stiffness matrix plays a crucial role....
Covariant Lyapunov vectors of chaotic Rayleigh-Bénard convection.
Xu, M; Paul, M R
2016-06-01
We explore numerically the high-dimensional spatiotemporal chaos of Rayleigh-Bénard convection using covariant Lyapunov vectors. We integrate the three-dimensional and time-dependent Boussinesq equations for a convection layer in a shallow square box geometry with an aspect ratio of 16 for very long times and for a range of Rayleigh numbers. We simultaneously integrate many copies of the tangent space equations in order to compute the covariant Lyapunov vectors. The dynamics explored has fractal dimensions of 20≲D_{λ}≲50, and we compute on the order of 150 covariant Lyapunov vectors. We use the covariant Lyapunov vectors to quantify the degree of hyperbolicity of the dynamics and the degree of Oseledets splitting and to explore the temporal and spatial dynamics of the Lyapunov vectors. Our results indicate that the chaotic dynamics of Rayleigh-Bénard convection is nonhyperbolic for all of the Rayleigh numbers we have explored. Our results yield that the entire spectrum of covariant Lyapunov vectors that we have computed are tangled as indicated by near tangencies with neighboring vectors. A closer look at the spatiotemporal features of the Lyapunov vectors suggests contributions from structures at two different length scales with differing amounts of localization.
Stability Analysis of MEMS Gyroscope Dynamic Systems
M. Naser-Moghadasi; S. A. Olamaei; F. Setoudeh
2013-01-01
In this paper, the existence of a common quadratic Lyapunov function for stability analysis of MEMS Gyroscope dynamic systems has been studied then a new method based on stochastic stability of MEMS Gyroscope system has been proposed.
Stability and robustness analysis of a linear time-periodic system subjected to random perturbations
Redkar, Sangram; Liu, J.; Sinha, S. C.
2012-03-01
In this work, new methods of guaranteeing the stability of linear time periodic dynamical systems with stochastic perturbations are presented. In the approaches presented here, the Lyapunov-Floquet (L-F) transformation is applied first so that the linear time-periodic part of the equations becomes time-invariant. For the linear time periodic system with stochastic perturbations, a stability theorem and related corollary have been suggested using the results previously obtained by Infante. This technique is not only applicable to systems with stochastic parameters but also to systems with deterministic variation in parameters. Some illustrative examples are presented to show the practical applications. These methods can be used to investigate the degree of robustness and design controllers for systems with time periodic coefficients subjected to random perturbations.
Transfinite Approximation of Hindman's Theorem
Beiglböck, Mathias
2010-01-01
Hindman's Theorem states that in any finite coloring of the integers, there is an infinite set all of whose finite sums belong to the same color. This is much stronger than the corresponding finite form, stating that in any finite coloring of the integers there are arbitrarily long finite sets with the same property. We extend the finite form of Hindman's Theorem to a "transfinite" version for each countable ordinal, and show that Hindman's Theorem is equivalent to the appropriate transfinite approximation holding for every countable ordinal. We then give a proof of Hindman's Theorem by directly proving these transfinite approximations.
General H-theorem and Entropies that Violate the Second Law
Directory of Open Access Journals (Sweden)
Alexander N. Gorban
2014-04-01
Full Text Available H-theorem states that the entropy production is nonnegative and, therefore, the entropy of a closed system should monotonically change in time. In information processing, the entropy production is positive for random transformation of signals (the information processing lemma. Originally, the H-theorem and the information processing lemma were proved for the classical Boltzmann-Gibbs-Shannon entropy and for the correspondent divergence (the relative entropy. Many new entropies and divergences have been proposed during last decades and for all of them the H-theorem is needed. This note proposes a simple and general criterion to check whether the H-theorem is valid for a convex divergence H and demonstrates that some of the popular divergences obey no H-theorem. We consider systems with n states Ai that obey first order kinetics (master equation. A convex function H is a Lyapunov function for all master equations with given equilibrium if and only if its conditional minima properly describe the equilibria of pair transitions Ai ⇌ Aj . This theorem does not depend on the principle of detailed balance and is valid for general Markov kinetics. Elementary analysis of pair equilibria demonstrate that the popular Bregman divergences like Euclidian distance or Itakura-Saito distance in the space of distribution cannot be the universal Lyapunov functions for the first-order kinetics and can increase in Markov processes. Therefore, they violate the second law and the information processing lemma. In particular, for these measures of information (divergences random manipulation with data may add information to data. The main results are extended to nonlinear generalized mass action law kinetic equations.
Controller design for TS models using delayed nonquadratic Lyapunov functions.
Lendek, Zsofia; Guerra, Thierry-Marie; Lauber, Jimmy
2015-03-01
In the last few years, nonquadratic Lyapunov functions have been more and more frequently used in the analysis and controller design for Takagi-Sugeno fuzzy models. In this paper, we developed relaxed conditions for controller design using nonquadratic Lyapunov functions and delayed controllers and give a general framework for the use of such Lyapunov functions. The two controller design methods developed in this framework outperform and generalize current state-of-the-art methods. The proposed methods are extended to robust and H∞ control and α -sample variation.
Lyapunov exponents for a Duffing oscillator
Zeni, Andrea R.; Gallas, Jason A. C.
With the help of a parallel computer we perform a systematic computation of Lyapunov exponents for a Duffing oscillator driven externally by a force proportional to cos( t). In contrast to the familiar situation in discrete-time systems where one finds “windows” of regularity embedded in intervals of chaos, we find the continuous-time Duffing oscillator to contain a quite regular epetition of relatively self-similar “islands of chaos” (i.e. regions characterized by positive exponents) embedded in large “seas of regularity” (negative exponents). We also investigate the effect of driving the oscillator with a Jacobian elliptic function cn( t, m). For m = 0 one has cn( t, 0) ≡ cos( t), the usual trigonometric pumping. For m = 1 one has cn( t, 1) ≡ sech( t), a hyperbolic pumping. When 0 displace the islands of chaos in parameter space. Thus, Jacobian pumping provides a possible way of “cleaning chaos” in regions of the parameter space for periodically driven systems.
Institute of Scientific and Technical Information of China (English)
郑秀亮; 柳洁冰
2016-01-01
研究了一类连续型时滞与离散型变时滞的非自治n种群Gilpin-Ayala型竞争系统,分别利用比较原理和构造Lyapunov函数得到了系统持久生存与全局渐近稳定性的充分条件,通过数值举例验证了定理的可行性.%In this paper, a class of nonautonomous n species Gilpin-Ayala competition system with continuous delays and discrete varying delays is studied, respectively, the sufficient conditions are obtained for the permanence and global asymp-totic stability of the system by using the comparison theorem and constructing a suitable Lyapunov functional, theorems are realized by giving a numerical example.
Universal scaling of Lyapunov-exponent fluctuations in space-time chaos.
Pazó, Diego; López, Juan M; Politi, Antonio
2013-06-01
Finite-time Lyapunov exponents of generic chaotic dynamical systems fluctuate in time. These fluctuations are due to the different degree of stability across the accessible phase space. A recent numerical study of spatially extended systems has revealed that the diffusion coefficient D of the Lyapunov exponents (LEs) exhibits a nontrivial scaling behavior, D(L)~L(-γ), with the system size L. Here, we show that the wandering exponent γ can be expressed in terms of the roughening exponents associated with the corresponding "Lyapunov surface." Our theoretical predictions are supported by the numerical analysis of several spatially extended systems. In particular, we find that the wandering exponent of the first LE is universal: in view of the known relationship with the Kardar-Parisi-Zhang equation, γ can be expressed in terms of known critical exponents. Furthermore, our simulations reveal that the bulk of the spectrum exhibits a clearly different behavior and suggest that it belongs to a possibly unique universality class, which has, however, yet to be identified.
Convex Optimization methods for computing the Lyapunov Exponent of matrices
Protasov, Vladimir Yu
2012-01-01
We introduce a new approach to evaluate the largest Lyapunov exponent of a family of nonnegative matrices. The method is based on using special positive homogeneous functionals on $R^{d}_+,$ which gives iterative lower and upper bounds for the Lyapunov exponent. They improve previously known bounds and converge to the real value. The rate of convergence is estimated and the efficiency of the algorithm is demonstrated on several problems from applications (in functional analysis, combinatorics, and lan- guage theory) and on numerical examples with randomly generated matrices. The method computes the Lyapunov exponent with a prescribed accuracy in relatively high dimensions (up to 60). We generalize this approach to all matrices, not necessar- ily nonnegative, derive a new universal upper bound for the Lyapunov exponent, and show that such a lower bound, in general, does not exist.
The Non-Signalling theorem in generalizations of Bell's theorem
Walleczek, J.; Grössing, G.
2014-04-01
Does "epistemic non-signalling" ensure the peaceful coexistence of special relativity and quantum nonlocality? The possibility of an affirmative answer is of great importance to deterministic approaches to quantum mechanics given recent developments towards generalizations of Bell's theorem. By generalizations of Bell's theorem we here mean efforts that seek to demonstrate the impossibility of any deterministic theories to obey the predictions of Bell's theorem, including not only local hidden-variables theories (LHVTs) but, critically, of nonlocal hidden-variables theories (NHVTs) also, such as de Broglie-Bohm theory. Naturally, in light of the well-established experimental findings from quantum physics, whether or not a deterministic approach to quantum mechanics, including an emergent quantum mechanics, is logically possible, depends on compatibility with the predictions of Bell's theorem. With respect to deterministic NHVTs, recent attempts to generalize Bell's theorem have claimed the impossibility of any such approaches to quantum mechanics. The present work offers arguments showing why such efforts towards generalization may fall short of their stated goal. In particular, we challenge the validity of the use of the non-signalling theorem as a conclusive argument in favor of the existence of free randomness, and therefore reject the use of the non-signalling theorem as an argument against the logical possibility of deterministic approaches. We here offer two distinct counter-arguments in support of the possibility of deterministic NHVTs: one argument exposes the circularity of the reasoning which is employed in recent claims, and a second argument is based on the inconclusive metaphysical status of the non-signalling theorem itself. We proceed by presenting an entirely informal treatment of key physical and metaphysical assumptions, and of their interrelationship, in attempts seeking to generalize Bell's theorem on the basis of an ontic, foundational
A Time scales Noether's theorem
Anerot, Baptiste; Cresson, Jacky; Pierret, Frédéric
2016-01-01
We prove a time scales version of the Noether's theorem relating group of symmetries and conservation laws. Our result extends the continuous version of the Noether's theorem as well as the discrete one and corrects a previous statement of Bartosiewicz and Torres in \\cite{BT}.
Abelian theorems for Whittaker transforms
Directory of Open Access Journals (Sweden)
Richard D. Carmichael
1987-01-01
Full Text Available Initial and final value Abelian theorems for the Whittaker transform of functions and of distributions are obtained. The Abelian theorems are obtained as the complex variable of the transform approaches 0 or ∞ in absolute value inside a wedge region in the right half plane.
Lopez-Real, Francis
2008-01-01
While the author was searching the web, he came across an article by Michael Keyton of IMSA (Illinois Mathematics and Science Academy) called "Theorems of mystery". The phrase is Keyton's own, and he defines such a theorem as "a result that has considerable structure with minimal hypotheses." The simplest of his 10 examples is one that many…
Directory of Open Access Journals (Sweden)
Coghetto Roland
2015-06-01
Full Text Available Morley’s trisector theorem states that “The points of intersection of the adjacent trisectors of the angles of any triangle are the vertices of an equilateral triangle” [10]. There are many proofs of Morley’s trisector theorem [12, 16, 9, 13, 8, 20, 3, 18]. We follow the proof given by A. Letac in [15].
Geometry of the Adiabatic Theorem
Lobo, Augusto Cesar; Ribeiro, Rafael Antunes; Ribeiro, Clyffe de Assis; Dieguez, Pedro Ruas
2012-01-01
We present a simple and pedagogical derivation of the quantum adiabatic theorem for two-level systems (a single qubit) based on geometrical structures of quantum mechanics developed by Anandan and Aharonov, among others. We have chosen to use only the minimum geometric structure needed for the understanding of the adiabatic theorem for this case.…
Lyapunov control of quantum systems with impulsive control fields.
Yang, Wei; Sun, Jitao
2013-01-01
We investigate the Lyapunov control of finite-dimensional quantum systems with impulsive control fields, where the studied quantum systems are governed by the Schrödinger equation. By three different Lyapunov functions and the invariant principle of impulsive systems, we study the convergence of quantum systems with impulsive control fields and propose new results for the mentioned quantum systems in the form of sufficient conditions. Two numerical simulations are presented to illustrate the effectiveness of the proposed control method.
Passivity/Lyapunov based controller design for trajectory tracking of flexible joint manipulators
Sicard, Pierre; Wen, John T.; Lanari, Leonardo
1992-01-01
A passivity and Lyapunov based approach for the control design for the trajectory tracking problem of flexible joint robots is presented. The basic structure of the proposed controller is the sum of a model-based feedforward and a model-independent feedback. Feedforward selection and solution is analyzed for a general model for flexible joints, and for more specific and practical model structures. Passivity theory is used to design a motor state-based controller in order to input-output stabilize the error system formed by the feedforward. Observability conditions for asymptotic stability are stated and verified. In order to accommodate for modeling uncertainties and to allow for the implementation of a simplified feedforward compensation, the stability of the system is analyzed in presence of approximations in the feedforward by using a Lyapunov based robustness analysis. It is shown that under certain conditions, e.g., the desired trajectory is varying slowly enough, stability is maintained for various approximations of a canonical feedforward.
Some Theorems on Generalized Basic Hypergeometric Series
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A. D. Wadhwa
1972-07-01
Full Text Available In an earlier paper the author has established two theorems on generalized hypergeometric functions. In each theorem a numerator differs from a denominator by a positive integer. These theorems were further used to prove some theorems on the sums of Kampe de Feriet functions. Here, we have established the theorems which are the basic analogues of the theorems proved in the earlier paper.
Combinatorial Reciprocity Theorems
Beck, Matthias
2012-01-01
A common theme of enumerative combinatorics is formed by counting functions that are polynomials evaluated at positive integers. In this expository paper, we focus on four families of such counting functions connected to hyperplane arrangements, lattice points in polyhedra, proper colorings of graphs, and $P$-partitions. We will see that in each instance we get interesting information out of a counting function when we evaluate it at a \\emph{negative} integer (and so, a priori the counting function does not make sense at this number). Our goals are to convey some of the charm these "alternative" evaluations of counting functions exhibit, and to weave a unifying thread through various combinatorial reciprocity theorems by looking at them through the lens of geometry, which will include some scenic detours through other combinatorial concepts.
Towards the Carpenter's Theorem
Argerami, Martin
2008-01-01
Let M be a II_1 factor, A a masa in M and E the unique conditional expectation on A. Under some technical assumptions on the inclusion of A in M, which hold true for any semiregular masa of a separable factor, we show that for every discrete a in the positive part of the unit ball of A it is possible to find a projection p in M such that E(p)=a$. We also show an example of a diffuse operator x in A such that there exists a projection q in M with E(q)=x. These results show a new family of instances of a conjecture by Kadison, the so-called ``Carpenter's Theorem''.
Vector Lyapunov Functions for Stochastic Interconnected Systems
Boussalis, D.
1985-01-01
Theoretical paper presents set of sufficient conditions for asymptotic and exponential stability with probability 1 for class of stochastic interconnected systems. Theory applicable to complicated, large-scale mechanical or electrical systems, and, for several design problems, it reduces computational difficulty by relating stability criteria to fundamental structural features of system.
Vector Lyapunov Functions for Stochastic Interconnected Systems
Boussalis, D.
1985-01-01
Theoretical paper presents set of sufficient conditions for asymptotic and exponential stability with probability 1 for class of stochastic interconnected systems. Theory applicable to complicated, large-scale mechanical or electrical systems, and, for several design problems, it reduces computational difficulty by relating stability criteria to fundamental structural features of system.
Institute of Scientific and Technical Information of China (English)
邓飞其; 赵碧蓉; 罗琦
2005-01-01
Based on stochastic Fubini theorem,the Hopfield neural network system depicted by a stochastic partial differential equation is translated into a stochastic ordinary differential equation. By constructing a mean Lyapunov function with respect to the space variables and using Ito formula under the integral operators, the exponential stability of stochastic neutral systems with distributed parameters is investigated by deviating of the function along the trajectories of the systems. Also, the Lyapunov exponent estimate is obtained. Thus, the stability of stochastic systems with distributed parameters is studied by Lyapunov direct method.%基于随机Fubini定理,将随机偏微分方程描述的Hopfield神经网络系统转化为用相应的随机常微分方程来描述.利用关于空间变量平均的Lyapunov函数与Ito公式,通过对所构造的Lyapunov函数在Ito微分规则下对相应系统求导的方法,获得了系统指数稳定的代数判据及其Lyapunov指数估计.实现了运用Lyapunov直接法对分布参数系统稳定性的研究.
Smorynski, Craig
2017-01-01
This book is about the rise and supposed fall of the mean value theorem. It discusses the evolution of the theorem and the concepts behind it, how the theorem relates to other fundamental results in calculus, and modern re-evaluations of its role in the standard calculus course. The mean value theorem is one of the central results of calculus. It was called “the fundamental theorem of the differential calculus” because of its power to provide simple and rigorous proofs of basic results encountered in a first-year course in calculus. In mathematical terms, the book is a thorough treatment of this theorem and some related results in the field; in historical terms, it is not a history of calculus or mathematics, but a case study in both. MVT: A Most Valuable Theorem is aimed at those who teach calculus, especially those setting out to do so for the first time. It is also accessible to anyone who has finished the first semester of the standard course in the subject and will be of interest to undergraduate mat...
Are Bred Vectors The Same As Lyapunov Vectors?
Kalnay, E.; Corazza, M.; Cai, M.
Regional loss of predictability is an indication of the instability of the underlying flow, where small errors in the initial conditions (or imperfections in the model) grow to large amplitudes in finite times. The stability properties of evolving flows have been studied using Lyapunov vectors (e.g., Alligood et al, 1996, Ott, 1993, Kalnay, 2002), singular vectors (e.g., Lorenz, 1965, Farrell, 1988, Molteni and Palmer, 1993), and, more recently, with bred vectors (e.g., Szunyogh et al, 1997, Cai et al, 2001). Bred vectors (BVs) are, by construction, closely related to Lyapunov vectors (LVs). In fact, after an infinitely long breeding time, and with the use of infinitesimal ampli- tudes, bred vectors are identical to leading Lyapunov vectors. In practical applications, however, bred vectors are different from Lyapunov vectors in two important ways: a) bred vectors are never globally orthogonalized and are intrinsically local in space and time, and b) they are finite-amplitude, finite-time vectors. These two differences are very significant in a dynamical system whose size is very large. For example, the at- mosphere is large enough to have "room" for several synoptic scale instabilities (e.g., storms) to develop independently in different regions (say, North America and Aus- tralia), and it is complex enough to have several different possible types of instabilities (such as barotropic, baroclinic, convective, and even Brownian motion). Bred vectors share some of their properties with leading LVs (Corazza et al, 2001a, 2001b, Toth and Kalnay, 1993, 1997, Cai et al, 2001). For example, 1) Bred vectors are independent of the norm used to define the size of the perturba- tion. Corazza et al. (2001) showed that bred vectors obtained using a potential enstro- phy norm were indistinguishable from bred vectors obtained using a streamfunction squared norm, in contrast with singular vectors. 2) Bred vectors are independent of the length of the rescaling period as long as the
随机逼近中的Lyapunov函数%On Lyapunov Functions inStochastic Approximation
Institute of Scientific and Technical Information of China (English)
张俊华
2001-01-01
本文研究了随机逼近中满足某种条件的Lyapunov函数的存在性及如何构造Lyapunov函数的问题,讨论了算法收敛性与相应常微分方程系统的渐近稳定性之间的关系.%In this paper, we investigate existence and construction of certain Lyapunov functions instochastic approximation (SA) and discuss the relation between convergence of SA algorithms andasymptotic stability of the corresponding ordinary differential equation systems.
Splitting Spacetime and Cloning Qubits: Linking No-Go Theorems across the ER=EPR Duality
Bao, Ning; Remmen, Grant N
2015-01-01
We analyze the no-cloning theorem in quantum mechanics through the lens of the proposed ER=EPR (Einstein-Rosen = Einstein-Podolsky-Rosen) duality between entanglement and wormholes. In particular, we find that the no-cloning theorem is dual on the gravity side to the no-go theorem for topology change, violating the axioms of which allows for wormhole stabilization and causality violation. Such a duality between important no-go theorems elucidates the proposed connection between spacetime geometry and quantum entanglement.
Splitting spacetime and cloning qubits: linking no-go theorems across the ER=EPR duality
Energy Technology Data Exchange (ETDEWEB)
Bao, Ning [Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125 (United States); Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125 (United States); Pollack, Jason; Remmen, Grant N. [Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125 (United States)
2015-11-15
We analyze the no-cloning theorem in quantum mechanics through the lens of the proposed ER=EPR (Einstein-Rosen = Einstein-Podolsky-Rosen) duality between entanglement and wormholes. In particular, we find that the no-cloning theorem is dual on the gravity side to the no-go theorem for topology change, violating the axioms of which allows for wormhole stabilization and causality violation. Such a duality between important no-go theorems elucidates the proposed connection between spacetime geometry and quantum entanglement. (copyright 2015 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim)
Lyapunov, Floquet, and singular vectors for baroclinic waves
Directory of Open Access Journals (Sweden)
R. M. Samelson
2001-01-01
Full Text Available The dynamics of the growth of linear disturbances to a chaotic basic state is analyzed in an asymptotic model of weakly nonlinear, baroclinic wave-mean interaction. In this model, an ordinary differential equation for the wave amplitude is coupled to a partial differential equation for the zonal flow correction. The leading Lyapunov vector is nearly parallel to the leading Floquet vector f1 of the lowest-order unstable periodic orbit over most of the attractor. Departures of the Lyapunov vector from this orientation are primarily rotations of the vector in an approximate tangent plane to the large-scale attractor structure. Exponential growth and decay rates of the Lyapunov vector during individual Poincaré section returns are an order of magnitude larger than the Lyapunov exponent l ≈ 0.016. Relatively large deviations of the Lyapunov vector from parallel to f1 are generally associated with relatively large transient decays. The transient growth and decay of the Lyapunov vector is well described by the transient growth and decay of the leading Floquet vectors of the set of unstable periodic orbits associated with the attractor. Each of these vectors is also nearly parallel to f1. The dynamical splitting of the complete sets of Floquet vectors for the higher-order cycles follows the previous results on the lowest-order cycle, with the vectors divided into wave-dynamical and decaying zonal flow modes. Singular vectors and singular values also generally follow this split. The primary difference between the leading Lyapunov and singular vectors is the contribution of decaying, inviscidly-damped wave-dynamical structures to the singular vectors.
-Dimensional Fractional Lagrange's Inversion Theorem
Directory of Open Access Journals (Sweden)
F. A. Abd El-Salam
2013-01-01
Full Text Available Using Riemann-Liouville fractional differential operator, a fractional extension of the Lagrange inversion theorem and related formulas are developed. The required basic definitions, lemmas, and theorems in the fractional calculus are presented. A fractional form of Lagrange's expansion for one implicitly defined independent variable is obtained. Then, a fractional version of Lagrange's expansion in more than one unknown function is generalized. For extending the treatment in higher dimensions, some relevant vectors and tensors definitions and notations are presented. A fractional Taylor expansion of a function of -dimensional polyadics is derived. A fractional -dimensional Lagrange inversion theorem is proved.
Complex integration and Cauchy's theorem
Watson, GN
2012-01-01
This brief monograph by one of the great mathematicians of the early twentieth century offers a single-volume compilation of propositions employed in proofs of Cauchy's theorem. Developing an arithmetical basis that avoids geometrical intuitions, Watson also provides a brief account of the various applications of the theorem to the evaluation of definite integrals.Author G. N. Watson begins by reviewing various propositions of Poincaré's Analysis Situs, upon which proof of the theorem's most general form depends. Subsequent chapters examine the calculus of residues, calculus optimization, the
Fluctuation theorem: A critical review
Malek Mansour, M.; Baras, F.
2017-10-01
Fluctuation theorem for entropy production is revisited in the framework of stochastic processes. The applicability of the fluctuation theorem to physico-chemical systems and the resulting stochastic thermodynamics were analyzed. Some unexpected limitations are highlighted in the context of jump Markov processes. We have shown that these limitations handicap the ability of the resulting stochastic thermodynamics to correctly describe the state of non-equilibrium systems in terms of the thermodynamic properties of individual processes therein. Finally, we considered the case of diffusion processes and proved that the fluctuation theorem for entropy production becomes irrelevant at the stationary state in the case of one variable systems.
Limit theorem and uniqueness theorem of backward stochastic differential equations
Institute of Scientific and Technical Information of China (English)
JIANG; Long
2006-01-01
This paper establishes a limit theorem for solutions of backward stochastic differential equations (BSDEs). By this limit theorem, this paper proves that, under the standard assumption g(t,y,0)≡0, the generator g of a BSDE can be uniquely determined by the corresponding g-expectation εg; this paper also proves that if a filtration consistent expectation ε can be represented as a g-expectation εg, then the corresponding generator g must be unique.
Cardioids and Morley's Trisector Theorem
J. Brinkhuis (Jan); van de Craats, J.
2017-01-01
textabstractA self-contained account of Morley's own proof of his celebrated trisector theorem is given. This makes this elegant and almost forgotten fragment of analytic Euclidean geometry more accessible to modern readers
Statistics, Causality and Bell's theorem
Gill, Richard D
2012-01-01
Bell's (1964) theorem is popularly supposed to establish the non-locality of quantum physics as a mathematical-physical theory. Building from this, observed violation of Bell's inequality in experiments such as that of Aspect and coworkers (1982) is popularly supposed to provide empirical proof of non-locality in the real world. This paper reviews recent work on Bell's theorem, linking it to issues in causality as understood by statisticians. The paper starts with a new proof of a strong (finite sample) version of Bell's theorem which relies only on elementary arithmetic and (counting) probability. This proof underscores the fact that Bell's theorem tells us that quantum theory is incompatible with the conjunction of three cherished and formerly uncontroversial physical principles, nicknamed here locality, realism, and freedom. The first, locality, is obviously connected to causality: causal influences need time to propagate spatially. Less obviously, the other two principles, realism and freedom, are also fo...
Design of Optimal PID Controller with ɛ-Routh Stability for Different Processes
Directory of Open Access Journals (Sweden)
XianHong Li
2013-01-01
Full Text Available This paper presents a design method of the optimal proportional-integral-derivative (PID controller with ɛ-Routh stability for different processes through Lyapunov approach. The optimal PID controller could be acquired by minimizing an augmented integral squared error (AISE performance index which contains control error and at least first-order error derivative, or even may contain nth-order error derivative. The optimal control problem could be transformed into a nonlinear constraint optimization (NLCO problem via Lyapunov theorems. Therefore, optimal PID controller could be obtained by solving NLCO problem through interior method or other optimization methods. The proposed method can be applied for different processes, and optimal PID controllers under various control weight matrices and ɛ-Routh stability are presented for different processes. Control weight matrix and ɛ-Routh stability’s effects on system performances are studied, and different tuning methods’ system performances are also discussed. ɛ-Routh stability’s effects on disturbance rejection ability are investigated, and different tuning methods’ disturbances rejection ability is studied. To further illustrate the proposed method, experimental results of coupled water tank system (CWTS under different set points are presented. Both simulation results and experiment results show the effectiveness and usefulness of the proposed method.
The Kolmogorov-Riesz compactness theorem
Hanche-Olsen, Harald
2009-01-01
We show that the Arzela-Ascoli theorem and Kolmogorov compactness theorem both are consequences of a simple lemma on compactness in metric spaces. Their relation to Helly's theorem is discussed. The paper contains a detailed discussion on the historical background of the Kolmogorov compactness theorem.
The Second Noether Theorem on Time Scales
Directory of Open Access Journals (Sweden)
Agnieszka B. Malinowska
2013-01-01
Full Text Available We extend the second Noether theorem to variational problems on time scales. As corollaries we obtain the classical second Noether theorem, the second Noether theorem for the h-calculus and the second Noether theorem for the q-calculus.
The second Noether theorem on time scale
Malinowska, Agnieszka B.; Martins, Natália
2014-01-01
We extend the second Noether theorem to variational problems on time scales. Our result provides as corollaries the classical second Noether theorem, the second Noether theorem for the $h$-calculus and the second Noether theorem for the $q$-calculus.
Local virial and tensor theorems.
Cohen, Leon
2011-11-17
We show that for any wave function and potential the local virial theorem can always be satisfied 2K(r) = r·ΔV by choosing a particular expression for the local kinetic energy. In addition, we show that for each choice of local kinetic energy there are an infinite number of quasi-probability distributions which will generate the same expression. We also consider the local tensor virial theorem.
Directory of Open Access Journals (Sweden)
Quanqi Mu
2014-01-01
Full Text Available Compared with traditional mechanical inertially stabilized platform (ISP, magnetic suspension ISP (MSISP can absorb high frequency vibrations via a magnetic suspension bearing system with five degrees of freedom between azimuth and pitch gimbals. However, force acting between rotor and stator will introduce coupling torque to roll and pitch gimbals. Since the disturbance of magnetic bearings has strong nonlinearity, classic state feedback control algorithm cannot bring higher precision control for MSISP. In order to enhance the control accuracy for MSISP, a disturbance compensator based on radial basis function neural network (RBFNN is developed to compensate for the disturbance. Using the Lyapunov theorem, the weighting matrix of RBFNN can be updated online. Therefore, the RBFNN can be constructed without priori training. At last, simulations and experiment results validate that the compensation method proposed in this paper can improve ISP accuracy significantly.
Lyapunov exponent in quantum mechanics A phase-space approach
Man'ko, V I
2000-01-01
Using the symplectic tomography map, both for the probability distributionsin classical phase space and for the Wigner functions of its quantumcounterpart, we discuss a notion of Lyapunov exponent for quantum dynamics.Because the marginal distributions, obtained by the tomography map, are alwayswell defined probabilities, the correspondence between classical and quantumnotions is very clear. Then we also obtain the corresponding expressions inHilbert space. Some examples are worked out. Classical and quantum exponentsare seen to coincide for local and non-local time-dependent quadraticpotentials. For non-quadratic potentials classical and quantum exponents aredifferent and some insight is obtained on the taming effect of quantummechanics on classical chaos. A detailed analysis is made for the standard map.Providing an unambiguous extension of the notion of Lyapunov exponent toquantum mechnics, the method that is developed is also computationallyefficient in obtaining analytical results for the Lyapunov expone...
On Controllability and Observability of Fuzzy Dynamical Matrix Lyapunov Systems
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M. S. N. Murty
2008-04-01
Full Text Available We provide a way to combine matrix Lyapunov systems with fuzzy rules to form a new fuzzy system called fuzzy dynamical matrix Lyapunov system, which can be regarded as a new approach to intelligent control. First, we study the controllability property of the fuzzy dynamical matrix Lyapunov system and provide a sufficient condition for its controllability with the use of fuzzy rule base. The significance of our result is that given a deterministic system and a fuzzy state with rule base, we can determine the rule base for the control. Further, we discuss the concept of observability and give a sufficient condition for the system to be observable. The advantage of our result is that we can determine the rule base for the initial value without solving the system.
Detecting Epileptic Seizure from Scalp EEG Using Lyapunov Spectrum
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Truong Quang Dang Khoa
2012-01-01
Full Text Available One of the inherent weaknesses of the EEG signal processing is noises and artifacts. To overcome it, some methods for prediction of epilepsy recently reported in the literature are based on the evaluation of chaotic behavior of intracranial electroencephalographic (EEG recordings. These methods reduced noises, but they were hazardous to patients. In this study, we propose using Lyapunov spectrum to filter noise and detect epilepsy on scalp EEG signals only. We determined that the Lyapunov spectrum can be considered as the most expected method to evaluate chaotic behavior of scalp EEG recordings and to be robust within noises. Obtained results are compared to the independent component analysis (ICA and largest Lyapunov exponent. The results of detecting epilepsy are compared to diagnosis from medical doctors in case of typical general epilepsy.
Detecting epileptic seizure from scalp EEG using Lyapunov spectrum.
Khoa, Truong Quang Dang; Huong, Nguyen Thi Minh; Toi, Vo Van
2012-01-01
One of the inherent weaknesses of the EEG signal processing is noises and artifacts. To overcome it, some methods for prediction of epilepsy recently reported in the literature are based on the evaluation of chaotic behavior of intracranial electroencephalographic (EEG) recordings. These methods reduced noises, but they were hazardous to patients. In this study, we propose using Lyapunov spectrum to filter noise and detect epilepsy on scalp EEG signals only. We determined that the Lyapunov spectrum can be considered as the most expected method to evaluate chaotic behavior of scalp EEG recordings and to be robust within noises. Obtained results are compared to the independent component analysis (ICA) and largest Lyapunov exponent. The results of detecting epilepsy are compared to diagnosis from medical doctors in case of typical general epilepsy.
Characterizing weak chaos using time series of Lyapunov exponents.
da Silva, R M; Manchein, C; Beims, M W; Altmann, E G
2015-06-01
We investigate chaos in mixed-phase-space Hamiltonian systems using time series of the finite-time Lyapunov exponents. The methodology we propose uses the number of Lyapunov exponents close to zero to define regimes of ordered (stickiness), semiordered (or semichaotic), and strongly chaotic motion. The dynamics is then investigated looking at the consecutive time spent in each regime, the transition between different regimes, and the regions in the phase space associated to them. Applying our methodology to a chain of coupled standard maps we obtain (i) that it allows for an improved numerical characterization of stickiness in high-dimensional Hamiltonian systems, when compared to the previous analyses based on the distribution of recurrence times; (ii) that the transition probabilities between different regimes are determined by the phase-space volume associated to the corresponding regions; and (iii) the dependence of the Lyapunov exponents with the coupling strength.
Lyapunov exponents for synchronous 12-lead ECG signals
Institute of Scientific and Technical Information of China (English)
无
2002-01-01
The Lyapunov exponents of synchronous 12-lead ECG signals have been investigated for the first time using a multi-sensor (electrode) technique. The results show that the Lyapunov exponents computed from different locations on the body surface are not the same, but have a distribution characteristic for the ECG signals recorded from coronary artery disease (CAD) patients with sinus rhythms and for signals from healthy older people. The maximum Lyapunov exponent L1 of all signals is positive. While all the others are negative, so the ECG signal has chaotic characteristics. With the same leads, L1 of CAD patients is less than that of healthy people, so the CAD patients and healthy people can be classified by L1, L1 therefore has potential values in the diagnosis of heart disease.
Lyapunov exponents of stochastic systems—from micro to macro
Laffargue, Tanguy; Tailleur, Julien; van Wijland, Frédéric
2016-03-01
Lyapunov exponents of dynamical systems are defined from the rates of divergence of nearby trajectories. For stochastic systems, one typically assumes that these trajectories are generated under the ‘same noise realization’. The purpose of this work is to critically examine what this expression means. For Brownian particles, we consider two natural interpretations of the noise: intrinsic to the particles or stemming from the fluctuations of the environment. We show how they lead to different distributions of the largest Lyapunov exponent as well as different fluctuating hydrodynamics for the collective density field. We discuss, both at microscopic and macroscopic levels, the limits in which these noise prescriptions become equivalent. We close this paper by providing an estimate of the largest Lyapunov exponent and of its fluctuations for interacting particles evolving with Dean-Kawasaki dynamics.
Fixed-point-like theorems on subspaces
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Bernard Cornet
2004-08-01
Full Text Available We prove a fixed-point-like theorem for multivalued mappings defined on the finite Cartesian product of Grassmannian manifolds and convex sets. Our result generalizes two different kinds of theorems: the fixed-point-like theorem by Hirsch et al. (1990 or Husseini et al. (1990 and the fixed-point theorem by Gale and Mas-Colell (1975 (which generalizes Kakutani's theorem (1941.
Taming systematic uncertainties at the LHC with the central limit theorem
Energy Technology Data Exchange (ETDEWEB)
Fichet, Sylvain, E-mail: sylvain@ift.unesp.br
2016-10-15
We study the simplifications occurring in any likelihood function in the presence of a large number of small systematic uncertainties. We find that the marginalisation of these uncertainties can be done analytically by means of second-order error propagation, error combination, the Lyapunov central limit theorem, and under mild approximations which are typically satisfied for LHC likelihoods. The outcomes of this analysis are i) a very light treatment of systematic uncertainties ii) a convenient way of reporting the main effects of systematic uncertainties, such as the detector effects occurring in LHC measurements.
Taming systematic uncertainties at the LHC with the central limit theorem
Fichet, Sylvain
2016-01-01
We study the simplifications occurring in any likelihood function in the presence of a large number of small systematic uncertainties. We find that the marginalisation of these uncertainties can be done analytically by means of second-order error propagation, error combination, the Lyapunov central limit theorem, and under mild approximations which are typically satisfied for LHC likelihoods. The outcomes of this analysis are i) a very light treatment of systematic uncertainties ii) a convenient way of reporting the main effects of systematic uncertainties such as the detector effects occuring in LHC measurements.
RAZUMIKHIN-TYPE THEOREMS OF NEUTRAL STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS
Institute of Scientific and Technical Information of China (English)
Zhou Shaobo; Hu Shigeng
2009-01-01
The stability of stochastic functional differential equation with Markovian switching was studied by several authors, but there was almost no work on the stability of the neutral stochastic functional differential equations with Markovian switching. The aim of this article is to close this gap. The authors establish Razumikhin-type theorem of the neutral stochastic functional differential equations with Markovian switching, and those without Markovian switching.
Lyapunov based nonlinear control of electrical and mechanical systems
Behal, Aman
This Ph.D. dissertation describes the design and implementation of various control strategies centered around the following applications: (i) an improved indirect field oriented controller for the induction motor, (ii) partial state feedback control of an induction motor with saturation effects, (iii) tracking control of an underactuated surface vessel, and (iv) an attitude tracking controller for an underactuated spacecraft. The theory found in each of these sections is demonstrated through simulation or experimental results. An introduction to each of these four primary chapters can be found in chapter one. In the second chapter, the previously published tracking control of [16] 1 is presented in the indirect field oriented control (IFOC) notation to achieve exponential rotor velocity/rotor flux tracking. Specifically, it is illustrated how the proposed IFOC controller can be rewritten in the manner of [16] to allow for a direct Lyapunov stability proof. Experimental results (implemented with the IFOC algorithm) are provided to corroborate the efficacy of the algorithm. In the third chapter, a singularity-free, rotor position tracking controller is presented for the full order, nonlinear dynamic model of the induction motor that includes the effects of magnetic saturation. Specifically, by utilizing the pi-equivalent saturation model, an observer/controller strategy is designed that achieves semi-global exponential rotor position tracking and only requires stator current, rotor velocity, and rotor position measurements. Simulation and experimental results are included to demonstrate the efficacy of the proposed algorithm. In the fourth chapter, a continuous, time-varying tracking controller is designed that globally exponentially forces the position/orientation tracking error of an under-actuated surface vessel to a neighborhood about zero that can be made arbitrarily small (i.e., global uniformly ultimately boundedness (GUUB)). The result is facilitated by
Lyapunov inequalities for Partial Differential Equations at radial higher eigenvalues
Canada, Antonio
2011-01-01
This paper is devoted to the study of $L_{p}$ Lyapunov-type inequalities ($ \\ 1 \\leq p \\leq +\\infty$) for linear partial differential equations at radial higher eigenvalues. More precisely, we treat the case of Neumann boundary conditions on balls in $\\real^{N}$. It is proved that the relation between the quantities $p$ and $N/2$ plays a crucial role to obtain nontrivial and optimal Lyapunov inequalities. By using appropriate minimizing sequences and a detailed analysis about the number and distribution of zeros of radial nontrivial solutions, we show significant qualitative differences according to the studied case is subcritical, supercritical or critical.
Lyapunov exponents for multi-parameter tent and logistic maps.
McCartney, Mark
2011-12-01
The behaviour of logistic and tent maps is studied in cases where the control parameter is dependent on iteration number. Analytic results for global Lyapunov exponent are presented in the case of the tent map and numerical results are presented in the case of the logistic map. In the case of a tent map with N control parameters, the fraction of parameter space for which the global Lyapunov exponent is positive is calculated. The case of bi-parameter maps of period N are investigated.
An iterative decoupling solution method for large scale Lyapunov equations
Athay, T. M.; Sandell, N. R., Jr.
1976-01-01
A great deal of attention has been given to the numerical solution of the Lyapunov equation. A useful classification of the variety of solution techniques are the groupings of direct, transformation, and iterative methods. The paper summarizes those methods that are at least partly favorable numerically, giving special attention to two criteria: exploitation of a general sparse system matrix structure and efficiency in resolving the governing linear matrix equation for different matrices. An iterative decoupling solution method is proposed as a promising approach for solving large-scale Lyapunov equation when the system matrix exhibits a general sparse structure. A Fortran computer program that realizes the iterative decoupling algorithm is also discussed.
Do Finite-Size Lyapunov Exponents detect coherent structures?
Karrasch, Daniel; Haller, George
2013-12-01
Ridges of the Finite-Size Lyapunov Exponent (FSLE) field have been used as indicators of hyperbolic Lagrangian Coherent Structures (LCSs). A rigorous mathematical link between the FSLE and LCSs, however, has been missing. Here, we prove that an FSLE ridge satisfying certain conditions does signal a nearby ridge of some Finite-Time Lyapunov Exponent (FTLE) field, which in turn indicates a hyperbolic LCS under further conditions. Other FSLE ridges violating our conditions, however, are seen to be false positives for LCSs. We also find further limitations of the FSLE in Lagrangian coherence detection, including ill-posedness, artificial jump-discontinuities, and sensitivity with respect to the computational time step.
Lyapunov spectra of Coulombic and gravitational periodic systems
Kumar, Pankaj
2016-01-01
We compute Lyapunov spectra for Coulombic and gravitational versions of the one-dimensional systems of parallel sheets with periodic boundary conditions. Exact time evolution of tangent-space vectors are derived and are utilized toward computing Lypaunov characteristic exponents using an event-driven algorithm. The results indicate that the energy dependence of the largest Lyapunov exponent emulates that of Kolmogorov-entropy density for each system at different degrees of freedom. Our approach forms an effective and approximation-free tool toward studying the dynamical properties exhibited by the Coulombic and gravitational systems and finds applications in investigating indications of thermodynamic transitions in large versions of the spatially periodic systems.
An iterative decoupling solution method for large scale Lyapunov equations
Athay, T. M.; Sandell, N. R., Jr.
1976-01-01
A great deal of attention has been given to the numerical solution of the Lyapunov equation. A useful classification of the variety of solution techniques are the groupings of direct, transformation, and iterative methods. The paper summarizes those methods that are at least partly favorable numerically, giving special attention to two criteria: exploitation of a general sparse system matrix structure and efficiency in resolving the governing linear matrix equation for different matrices. An iterative decoupling solution method is proposed as a promising approach for solving large-scale Lyapunov equation when the system matrix exhibits a general sparse structure. A Fortran computer program that realizes the iterative decoupling algorithm is also discussed.
Bilinear Approximate Model-Based Robust Lyapunov Control for Parabolic Distributed Collectors
Elmetennani, Shahrazed
2016-11-09
This brief addresses the control problem of distributed parabolic solar collectors in order to maintain the field outlet temperature around a desired level. The objective is to design an efficient controller to force the outlet fluid temperature to track a set reference despite the unpredictable varying working conditions. In this brief, a bilinear model-based robust Lyapunov control is proposed to achieve the control objectives with robustness to the environmental changes. The bilinear model is a reduced order approximate representation of the solar collector, which is derived from the hyperbolic distributed equation describing the heat transport dynamics by means of a dynamical Gaussian interpolation. Using the bilinear approximate model, a robust control strategy is designed applying Lyapunov stability theory combined with a phenomenological representation of the system in order to stabilize the tracking error. On the basis of the error analysis, simulation results show good performance of the proposed controller, in terms of tracking accuracy and convergence time, with limited measurement even under unfavorable working conditions. Furthermore, the presented work is of interest for a large category of dynamical systems knowing that the solar collector is representative of physical systems involving transport phenomena constrained by unknown external disturbances.
Shah, Neerav
2011-01-01
The Magnetospheric MultiScale Mission (MMS) is scheduled to launch in late 2014. Its primary goal is to discover the fundamental plasma physics processes of reconnection in the Earth's magnetosphere. Each of the four MMS spacecraft is spin-stabilized at a nominal rate of 3 RPM. Traditional spin-stabilized spacecraft have used a number of separate modes to control nutation, spin rate, and precession. To reduce the number of modes and simplify operations, the Delta-H control mode is designed to accomplish nutation control, spin rate control, and precession control simultaneously. A nonlinear design technique, Lyapunov's method, is used to design the Delta-H control mode. A global spin rate controller selected as the baseline controller for MMS, proved to be insufficient due to an ambiguity in the attitude. Lyapunov's design method was used to solve this ambiguity, resulting in a controller that meets the design goals. Simulation results show the advantage of the pointing and rate controller for maneuvers larger than 90 deg and provide insight into the performance of this controller.
Frequency-domain L2-stability conditions for time-varying linear and nonlinear MIMO systems
Institute of Scientific and Technical Information of China (English)
Zhihong HUANG; Y. V. VENKATESH; Cheng XIANG; Tong Heng LEE
2014-01-01
The paper deals with the L2-stability analysis of multi-input-multi-output (MIMO) systems, governed by integral equations, with a matrix of periodic/aperiodic time-varying gains and a vector of monotone, non-monotone and quasi-monotone nonlin-earities. For nonlinear MIMO systems that are described by differential equations, most of the literature on stability is based on an application of quadratic forms as Lyapunov-function candidates. In contrast, a non-Lyapunov framework is employed here to derive new and more general L2-stability conditions in the frequency domain. These conditions have the following features:i) They are expressed in terms of the positive definiteness of the real part of matrices involving the transfer function of the linear time-invariant block and a matrix multiplier function that incorporates the minimax properties of the time-varying linear/nonlinear block. ii) For certain cases of the periodic time-varying gain, they contain, depending on the multiplier function chosen, no restrictions on the normalized rate of variation of the time-varying gain, but, for other periodic/aperiodic time-varying gains, they do. Overall, even when specialized to periodic-coefficient linear and nonlinear MIMO systems, the stability conditions are distinct from and less restrictive than recent results in the literature. No comparable results exist in the literature for aperiodic time-varying gains. Furthermore, some new stability results concerning the dwell-time problem and time-varying gain switching in linear and nonlinear MIMO systems with periodic/aperiodic matrix gains are also presented. Examples are given to illustrate a few of the stability theorems.
Nambu-Goldstone theorem and spin-statistics theorem
Fujikawa, Kazuo
2016-05-01
On December 19-21 in 2001, we organized a yearly workshop at Yukawa Institute for Theoretical Physics in Kyoto on the subject of “Fundamental Problems in Field Theory and their Implications”. Prof. Yoichiro Nambu attended this workshop and explained a necessary modification of the Nambu-Goldstone theorem when applied to non-relativistic systems. At the same workshop, I talked on a path integral formulation of the spin-statistics theorem. The present essay is on this memorable workshop, where I really enjoyed the discussions with Nambu, together with a short comment on the color freedom of quarks.
Global stabilization of nonlinear systems with uncertain structure
Institute of Scientific and Technical Information of China (English)
无
2005-01-01
The global stabilization problem of nonlinear systems with uncertain structure is dealt with. Based on control Lyapunov function (CLF), a sufficient and necessary condition for Lyapunov stabilization is given. From the condition,several simply sufficient conditions for the globally asymptotical stability are deduced. A state feedback control law is designed to globally asymptotically stabilize the equilibrium of the closed system. Last, a simulation shows the effectiveness of the method.
A conjecture on the norm of Lyapunov mapping
Institute of Scientific and Technical Information of China (English)
Daizhan CHENG; Yahong ZHU; Hongsheng QI
2009-01-01
A conjecture that the norm of Lyapunov mapping LA equals to its restriction to the symmetric set,S,i.e.,‖LA‖ = ‖LA |s‖ was proposed in [1].In this paper,a method for numerical testing is provided first.Then,some recent progress on this conjecture is presented.
Quadratic Lyapunov Function and Exponential Dichotomy on Time Scales
Institute of Scientific and Technical Information of China (English)
ZHANG JI; LIU ZHEN-XIN
2011-01-01
In this paper, we study the relationship between exponential dichotomy and quadratic Lyapunov function for the linear equation x△ ＝ A(t)x on time scales.Moreover, for the nonlinear perturbed equation x△ ＝ A(t)x + f(t,x) we give the instability of the zero solution when f is sufficiently small.
Alignment of Lyapunov Vectors: A Quantitative Criterion to Predict Catastrophes?
Beims, Marcus W; Gallas, Jason A C
2016-11-15
We argue that the alignment of Lyapunov vectors provides a quantitative criterion to predict catastrophes, i.e. the imminence of large-amplitude events in chaotic time-series of observables generated by sets of ordinary differential equations. Explicit predictions are reported for a Rössler oscillator and for a semiconductor laser with optoelectronic feedback.
Alignment of Lyapunov Vectors: A Quantitative Criterion to Predict Catastrophes?
Beims, Marcus W.; Gallas, Jason A. C.
2016-11-01
We argue that the alignment of Lyapunov vectors provides a quantitative criterion to predict catastrophes, i.e. the imminence of large-amplitude events in chaotic time-series of observables generated by sets of ordinary differential equations. Explicit predictions are reported for a Rössler oscillator and for a semiconductor laser with optoelectronic feedback.
The Lyapunov exponents of C~1 hyperbolic systems
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
Let f be a C 1 diffeomorphisim of smooth Riemannian manifold and preserve a hyperbolic ergodic measure μ. We prove that if the Osledec splitting is dominated, then the Lyapunov exponents of μ can be approximated by the exponents of atomic measures on hyperbolic periodic orbits.
Analysis of human standing balance by largest lyapunov exponent.
Liu, Kun; Wang, Hongrui; Xiao, Jinzhuang; Taha, Zahari
2015-01-01
The purpose of this research is to analyse the relationship between nonlinear dynamic character and individuals' standing balance by the largest Lyapunov exponent, which is regarded as a metric for assessing standing balance. According to previous study, the largest Lyapunov exponent from centre of pressure time series could not well quantify the human balance ability. In this research, two improvements were made. Firstly, an external stimulus was applied to feet in the form of continuous horizontal sinusoidal motion by a moving platform. Secondly, a multiaccelerometer subsystem was adopted. Twenty healthy volunteers participated in this experiment. A new metric, coordinated largest Lyapunov exponent was proposed, which reflected the relationship of body segments by integrating multidimensional largest Lyapunov exponent values. By using this metric in actual standing performance under sinusoidal stimulus, an obvious relationship between the new metric and the actual balance ability was found in the majority of the subjects. These results show that the sinusoidal stimulus can make human balance characteristics more obvious, which is beneficial to assess balance, and balance is determined by the ability of coordinating all body segments.
Lyapunov exponents and particle dispersion in drift wave turbulence
DEFF Research Database (Denmark)
Pedersen, T.S.; Michelsen, Poul; Juul Rasmussen, J.
1996-01-01
The Hasegawa-Wakatani model equations for resistive drift waves are solved numerically for a range of values of the coupling due to the parallel electron motion. The largest Lyapunov exponent, lambda(1), is calculated to quantify the unpredictability of the turbulent flow and compared to other...
From Lyapunov modes to their exponents for hard disk systems.
Chung, Tony; Truant, Daniel; Morriss, Gary P
2010-06-01
We demonstrate the preservation of the Lyapunov modes in a system of hard disks by the underlying tangent space dynamics. This result is exact for the Zero modes and correct to order ϵ for the Transverse and Longitudinal-Momentum modes, where ϵ is linear in the mode number. For sufficiently large mode numbers, the ϵ terms become significant and the dynamics no longer preserves the mode structure. We propose a modified Gram-Schmidt procedure based on orthogonality with respect to the center zero space that produces the exact numerical mode. This Gram-Schmidt procedure can also exploit the orthogonality between conjugate modes and their symplectic structure in order to find a simple relation that determines the Lyapunov exponent from the Lyapunov mode. This involves a reclassification of the modes into either direction preserving or form preserving. These analytic methods assume a knowledge of the ordering of the modes within the Lyapunov spectrum, but gives both predictive power for the values of the exponents from the modes and describes the modes in greater detail than was previously achievable. Thus the modes and the exponents contain the same information.
Lyapunov vectors and assimilation in the unstable subspace: theory and applications
Palatella, Luigi; Carrassi, Alberto; Trevisan, Anna
2013-06-01
Based on a limited number of noisy observations, estimation algorithms provide a complete description of the state of a system at current time. Estimation algorithms that go under the name of assimilation in the unstable subspace (AUS) exploit the nonlinear stability properties of the forecasting model in their formulation. Errors that grow due to sensitivity to initial conditions are efficiently removed by confining the analysis solution in the unstable and neutral subspace of the system, the subspace spanned by Lyapunov vectors with positive and zero exponents, while the observational noise does not disturb the system along the stable directions. The formulation of the AUS approach in the context of four-dimensional variational assimilation (4DVar-AUS) and the extended Kalman filter (EKF-AUS) and its application to chaotic models is reviewed. In both instances, the AUS algorithms are at least as efficient but simpler to implement and computationally less demanding than their original counterparts. As predicted by the theory when error dynamics is linear, the optimal subspace dimension for 4DVar-AUS is given by the number of positive and null Lyapunov exponents, while the EKF-AUS algorithm, using the same unstable and neutral subspace, recovers the solution of the full EKF algorithm, but dealing with error covariance matrices of a much smaller dimension and significantly reducing the computational burden. Examples of the application to a simplified model of the atmospheric circulation and to the optimal velocity model for traffic dynamics are given. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Lyapunov analysis: from dynamical systems theory to applications’.
Schubert, Sebastian
2015-01-01
One of the most relevant weather regimes in the mid latitudes atmosphere is the persistent deviation from the approximately zonally symmetric jet stream to the emergence of so-called blocking patterns. Such configurations are usually connected to exceptional local stability properties of the flow which come along with an improved local forecast skills during the phenomenon. It is instead extremely hard to predict onset and decay of blockings. Covariant Lyapunov Vectors (CLVs) offer a suitable characterization of the linear stability of a chaotic flow, since they represent the full tangent linear dynamics by a covariant basis which explores linear perturbations at all time scales. Therefore, we will test whether CLVs feature a signature of the blockings. We examine the CLVs for a quasi-geostrophic beta-plane two-layer model in a periodic channel baroclinically driven by a meridional temperature gradient $\\Delta T$. An orographic forcing enhances the emergence of localized blocked regimes. We detect the blockin...
Analysis of the Emergence in Swarm Model Based on Largest Lyapunov Exponent
Directory of Open Access Journals (Sweden)
Yu Wu
2011-01-01
Full Text Available Emergent behaviors of collective intelligence systems, exemplified by swarm model, have attracted broad interests in recent years. However, current research mostly stops at observational interpretations and qualitative descriptions of emergent phenomena and is essentially short of quantitative analysis and evaluation. In this paper, we conduct a quantitative study on the emergence of swarm model by using chaos analysis of complex dynamic systems. This helps to achieve a more exact understanding of emergent phenomena. In particular, we evaluate the emergent behaviors of swarm model quantitatively by using the chaos and stability analysis of swarm model based on largest Lyapunov exponent. It is concluded that swarm model is at the edge of chaos when emergence occurs, and whether chaotic or stable at the beginning, swarm model will converge to stability with the elapse of time along with interactions among agents.
Fluctuation theorems for quantum processes
Albash, Tameem; Marvian, Milad; Zanardi, Paolo
2013-01-01
We present fluctuation theorems and moment generating function equalities for generalized thermodynamic observables and quantum dynamics described by completely positive trace preserving (CPTP) maps, with and without feedback control. Our results include the quantum Jarzynski equality and Crooks fluctuation theorem, and clarify the special role played by the thermodynamic work and thermal equilibrium states in previous studies. We show that unitality replaces micro-reversibility as the condition for the physicality of the reverse process in our fluctuation theorems. We present an experimental application of our theory to the problem of extracting the system-bath coupling magnitude, which we do for a system of pairs of coupled superconducting flux qubits undergoing quantum annealing.
Lyapunov matrices approach to the parametric optimization of time-delay systems
Directory of Open Access Journals (Sweden)
Duda Józef
2015-09-01
Full Text Available In the paper a Lyapunov matrices approach to the parametric optimization problem of time-delay systems with a P-controller is presented. The value of integral quadratic performance index of quality is equal to the value of Lyapunov functional for the initial function of the time-delay system. The Lyapunov functional is determined by means of the Lyapunov matrix
Lyapunov Matrices Approach to the Parametric Optimization of a System with Two Delays
Directory of Open Access Journals (Sweden)
Duda Jozef
2016-09-01
Full Text Available In the paper a Lyapunov matrices approach to the parametric optimization problem of time-delay systems with two commensurate delays and a P-controller is presented. The value of integral quadratic performance index of quality is equal to the value of the Lyapunov functional for the initial function of time-delay system. The Lyapunov functional is determined by means of the Lyapunov matrix.
Improvement of Hartman's linearization theorem
Institute of Scientific and Technical Information of China (English)
SHI; Jinlin(史金麟)
2003-01-01
Hartman's linearization theorem tells us that if matrix A has no zero real part and f(x) isbounded and satisfies Lipchitz condition with small Lipchitzian constant, then there exists a homeomorphismof Rn sending the solutions of nonlinear system x' = Ax + f(x) onto the solutions of linear system x' = Ax.In this paper, some components of the nonlinear item f(x) are permitted to be unbounded and we provethe result of global topological linearization without any special limitation and adding any condition. Thus,Hartman's linearization theorem is improved essentially.
Poutiainen, H. (Hayley)
2015-01-01
Group theory is a mathematical domain where groups and their properties are studied. The evolution of group theory as an area of study is said to be the result of the parallel development of a variety of different studies in mathematics. Sylow’s Theorems were a set of theorems proved around the same time the concept of group theory was being established, in the 1870s. Sylow used permutation groups in his proofs which were then later generalized and shown to hold true for all finite groups. Th...
Louis M. Houston
2012-01-01
Sign data are the signs of signal added to noise. It is well known that a constant signal can be recovered from sign data. In this paper, we show that an integral over variant signal can be recovered from an integral over sign data based on the variant signal. We refer to this as a generalized sign data average. We use this result to derive a Green's theorem for sign data. Green's theorem is important to various seismic processing methods, including seismic migration. Results in this paper ge...
Noether theorems and higher derivatives
Townsend, Paul K
2016-01-01
A simple proof of Noether's first theorem involves the promotion of a constant symmetry parameter $\\epsilon$ to an arbitrary function of time; the Noether charge $Q$ is then the coefficient of $\\dot\\epsilon$ in the variation of the action. Here we examine the validity of this proof for Lagrangian mechanics with arbitrarily-high time derivatives, in which context "higher-level" analogs of Noether's theorem can be similarly proved, and "Noetherian charges" read off from, e.g. the coefficient of $\\ddot \\epsilon$ in the variation of the action. While $Q=0$ implies a restricted gauge invariance, an unrestricted gauge invariance requires zero Noetherian charges too. Some illustrative examples are considered.
Pasicki, Lech
2011-01-01
Many versions of the Stokes theorem are known. More advanced of them require complicated mathematical machinery to be formulated which discourages the users. Our theorem is sufficiently simple to suit the handbooks and yet it is pretty general, as we assume the differential form to be continuous on a compact set F(A) and C1 "inside" while F(A) is built of "bricks" and its inner part is a C1 manifold. There is no problem of orientability and the integrals under consideration are convergent. The proof is based on integration by parts and inner approximation.
Smith, Beth A.; Stergiou, Nicholas; Ulrich, Beverly D.
2010-01-01
In previous studies we found that while preadolescents with Down syndrome (DS) produce higher amounts of variability (Smith et al., 2007) and larger Lyapunov exponent (LyE) values (indicating more instability) during walking than peers with typical development (TD) (Buzzi & Ulrich, 2004), they also partition more of this into goal-equivalent variability (UCM//), that can be exploited to increase options for success when perturbed (Black et al., 2007). Here we use nonlinear methods to examine the patterns that characterize gait variability as it emerges, in toddlers with TD and with DS, rather than after years of practice. We calculated Lyapunov exponent (LyE) values to assess stability of leg trajectories. We also tested the use of 3 algorithms for surrogation analysis to investigate mathematical periodicity of toddlers’ strides. Results show that toddlers’ LyE values were not different between groups or with practice and strides of both groups become more periodic with practice. PMID:20237407
The truncated Second Main Theorem and uniqueness theorems
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
In this paper, we first establish a truncated Second Main Theorem for algebraically nondegenerate holomorphic mappings from the complex plane into a complex projective variety V intersecting hypersurfaces. We then prove some uniqueness results for meromorphic mappings. The result of Demailly about a partial solution to the Fujita’s conjecture is used.
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
The H∞ output feedback control problem for uncertain discrete-time switched systems is reasearched. A new characterization of stability and H∞ performance for the switched system under arbitrary switching is obtained by using switched Lyapunov function.Then,based on the characterization,a linear matrix inequality (LMI)approach is developed to design a switched output feedback controller which guarantees the stability and H∞ performance of the closed-loop system.A numerical example is presented to demonstrate the application of the proposed method.
Li, Hongfei; Jiang, Haijun; Hu, Cheng
2016-03-01
In this paper, we investigate a class of memristor-based BAM neural networks with time-varying delays. Under the framework of Filippov solutions, boundedness and ultimate boundedness of solutions of memristor-based BAM neural networks are guaranteed by Chain rule and inequalities technique. Moreover, a new method involving Yoshizawa-like theorem is favorably employed to acquire the existence of periodic solution. By applying the theory of set-valued maps and functional differential inclusions, an available Lyapunov functional and some new testable algebraic criteria are derived for ensuring the uniqueness and global exponential stability of periodic solution of memristor-based BAM neural networks. The obtained results expand and complement some previous work on memristor-based BAM neural networks. Finally, a numerical example is provided to show the applicability and effectiveness of our theoretical results.
Stability of Nonlinear Stochastic Discrete-Time Systems
2013-01-01
This paper studies the stability for nonlinear stochastic discrete-time systems. First of all, several definitions on stability are introduced, such as stability, asymptotical stability, and pth moment exponential stability. Moreover, using the method of the Lyapunov functionals, some efficient criteria for stochastic stability are obtained. Some examples are presented to illustrate the effectiveness of the proposed theoretical results.
Von Laue's theorem and its applications
Wang, Changbiao
2012-01-01
Von Laue's theorem is strictly proved in detail to clarify confusions in textbook and literature. This theorem is used to analyze the classical electron and the static electric field confined in a finite region of space.
The Classical Version of Stokes' Theorem Revisited
DEFF Research Database (Denmark)
Markvorsen, Steen
2005-01-01
Using only fairly simple and elementary considerations - essentially from first year undergraduate mathematics - we prove that the classical Stokes' theorem for any given surface and vector field in $\\mathbb{R}^{3}$ follows from an application of Gauss' divergence theorem to a suitable modification...... of the vector field in a tubular shell around the given surface. The intuitive appeal of the divergence theorem is thus applied to bootstrap a corresponding intuition for Stokes' theorem. The two stated classical theorems are (like the fundamental theorem of calculus) nothing but shadows of the general version...... to above. Our proof that Stokes' theorem follows from Gauss' divergence theorem goes via a well known and often used exercise, which simply relates the concepts of divergence and curl on the local differential level. The rest of the paper uses only integration in $1$, $2$, and $3$ variables together...
Lyapunov exponent for aging process in induction motor
Bayram, Duygu; Ünnü, Sezen Yıdırım; Şeker, Serhat
2012-09-01
Nonlinear systems like electrical circuits and systems, mechanics, optics and even incidents in nature may pass through various bifurcations and steady states like equilibrium point, periodic, quasi-periodic, chaotic states. Although chaotic phenomena are widely observed in physical systems, it can not be predicted because of the nature of the system. On the other hand, it is known that, chaos is strictly dependent on initial conditions of the system [1-3]. There are several methods in order to define the chaos. Phase portraits, Poincaré maps, Lyapunov Exponents are the most common techniques. Lyapunov Exponents are the theoretical indicator of the chaos, named after the Russian mathematician Aleksandr Lyapunov (1857-1918). Lyapunov Exponents stand for the average exponential divergence or convergence of nearby system states, meaning estimating the quantitive measure of the chaotic attractor. Negative numbers of the exponents stand for a stable system whereas zero stands for quasi-periodic systems. On the other hand, at least if one of the exponents is positive, this situation is an indicator of the chaos. For estimating the exponents, the system should be modeled by differential equation but even in that case mathematical calculation of Lyapunov Exponents are not very practical and evaluation of these values requires a long signal duration [4-7]. For experimental data sets, it is not always possible to acquire the differential equations. There are several different methods in literature for determining the Lyapunov Exponents of the system [4, 5]. Induction motors are the most important tools for many industrial processes because they are cheap, robust, efficient and reliable. In order to have healthy processes in industrial applications, the conditions of the machines should be monitored and the different working conditions should be addressed correctly. To the best of our knowledge, researches related to Lyapunov exponents and electrical motors are mostly
Shell theorem for spontaneous emission
DEFF Research Database (Denmark)
Kristensen, Philip Trøst; Mortensen, Jakob Egeberg; Lodahl, Peter;
2013-01-01
and therefore is given exactly by the dipole approximation theory. This surprising result is a spontaneous emission counterpart to the shell theorems of classical mechanics and electrostatics and provides insights into the physics of mesoscopic emitters as well as great simplifications in practical calculations....
JACKSON'S THEOREM FOR COMPACT GROUPS
Institute of Scientific and Technical Information of China (English)
H. Vaezi; S. F. Rzaev
2002-01-01
In this article we consider the generalized shift operator defined by(Sh.f)(g) = ∫Gf (tut-1g)dton compact group G and by help of this operator we define "Spherical" modulus of continuity. So we proveStechkin and Jackson type theorems.
LUROTH'S THEOREM IN DIFFERENTIAL FIELDS
Institute of Scientific and Technical Information of China (English)
GAO Xiaoshan; XU Tao
2002-01-01
In this paper, we present a constructive proof of Liroth's theorem in differentialcase. We also give a method to find the inversion maps for general differential rationalparametric equations. As a consequence, we prove that a differential rational curve alwayshas a set of proper parametric equations.
Angle Defect and Descartes' Theorem
Scott, Paul
2006-01-01
Rene Descartes lived from 1596 to 1650. His contributions to geometry are still remembered today in the terminology "Descartes' plane". This paper discusses a simple theorem of Descartes, which enables students to easily determine the number of vertices of almost every polyhedron. (Contains 1 table and 2 figures.)
GENERALIZED RECIPROCAL THEOREMS AND THEIR APPLICATIONS
Institute of Scientific and Technical Information of China (English)
付宝连
2002-01-01
Generalized reciprocal theorems of non-coupled and coupled systems , which are valid for two deformed bodies with different constitutive relations are established by generalizing the idea of Betti ' s reciprocal theorem. When the constitutive relations of the two deformed bodies are all alike and linear elastic, the generalized reciprocal theorem of non-coupled systems just becomes Betti' s . Meanwhile, the generalized reciprocal theorems are applied to simulate calculations in elasticity.
A definability theorem for first order logic
Butz, C.; Moerdijk, I.
2001-01-01
In this paper we will present a definability theorem for first order logic This theorem is very easy to state and its proof only uses elementary tools To explain the theorem let us first observe that if M is a model of a theory T in a language L then clearly any definable subset S M ie a subset S
Ordering in mechanical geometry theorem proving
Institute of Scientific and Technical Information of China (English)
李洪波￥
1997-01-01
Ordering in mechanical geometry theorem proving is studied from geometric viewpoint and some new ideas are proposed. For Thebault’s theorem which is the most difficult theorem that has ever been proved by Wu’ s method, a very simple proof using Wu’s method under a linear order is discovered.
A note on generalized Weyl's theorem
Zguitti, H.
2006-04-01
We prove that if either T or T* has the single-valued extension property, then the spectral mapping theorem holds for B-Weyl spectrum. If, moreover T is isoloid, and generalized Weyl's theorem holds for T, then generalized Weyl's theorem holds for f(T) for every . An application is given for algebraically paranormal operators.
Stochastic stability of mechanical systems under renewal jump process parametric excitation
DEFF Research Database (Denmark)
Iwankiewicz, R.; Nielsen, Søren R.K.; Larsen, Jesper Winther
2005-01-01
if the state space is augmented by the products of the original state variables and the excitation variable. Asymptotic mean and mean-square stability as well as asymptotic sample (Lyapunov) stability with probability 1 are investigated. The Lyapunov exponents have been evaluated both by the direct simulation...
STABILITY OF SOLUTIONS TO CERTAIN FOURTH-ORDER DELAY DIFFERENTIAL EQUATIONS
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
By the Lyapunov functional approach, some better results on the asymptotic stabiBy the Lyapunov functional approach, some better results on the asymptotic stability and global asymptotic stability of zero solution to a certain fourth-order non-linear differential equation with delay are obtained.
Institute of Scientific and Technical Information of China (English)
Xianming ZHANG; Min WU; Jinhua SHE; Dongsheng HAN
2007-01-01
This paper addresses the problems of the robust stability and robust stabilization of a discrete-time system with polytopic uncertainties.A new and simple method is presented to directly decouple the Lyapunov matrix and the system dynamic matrix.Combining this method with the parameter-dependent Lyapunov function approach yields new criteria that include some existing ones as special cases.A numerical example illustrates the improvement over the existing ones.
On finite-size Lyapunov exponents in multiscale systems
Mitchell, Lewis
2012-01-01
We study the effect of regime switches on finite size Lyapunov exponents (FSLEs) in determining the error growth rates and predictability of multiscale systems. We consider a dynamical system involving slow and fast regimes and switches between them. The surprising result is that due to the presence of regimes the error growth rate can be a non-monotonic function of initial error amplitude. In particular, troughs in the large scales of FSLE spectra is shown to be a signature of slow regimes, whereas fast regimes are shown to cause large peaks in the spectra where error growth rates far exceed those estimated from the maximal Lyapunov exponent. We present analytical results explaining these signatures and corroborate them with numerical simulations. We show further that these peaks disappear in stochastic parametrizations of the fast chaotic processes, and the associated FSLE spectra reveal that large scale predictability properties of the full deterministic model are well approximated whereas small scale feat...
Behavior of the Lyapunov Exponent and Phase Transition in Nuclei
Institute of Scientific and Technical Information of China (English)
WANG Nan; WU Xi-Zhen; LI Zhu-Xia; WANG Ning; ZHUO Yi-Zhong; SUN Xiu-Quan
2000-01-01
Based on the quantum molecular dynamics model, we investigate the dynamical behaviors of the excited nuclear system to simulate the latter stage of heavy ion reactions, which associate with a liquid-gas phase transition. We try to search a microscopic way to describe the phase transition in realnuclei. The Lyapunov exponent is employed and examined for our purpose. We find out that the Lyapunov exponent is one of good microscopic quantities to describe the phase transition in hot nuclei. Coulomb potential and the finite size effect may give a strong influence on the critical temperature. However, the collision term plays a minor role in the process of the liquid-gas phase transition in finite systems.
Lyapunov exponents a tool to explore complex dynamics
Pikovsky, Arkady
2016-01-01
Lyapunov exponents lie at the heart of chaos theory, and are widely used in studies of complex dynamics. Utilising a pragmatic, physical approach, this self-contained book provides a comprehensive description of the concept. Beginning with the basic properties and numerical methods, it then guides readers through to the most recent advances in applications to complex systems. Practical algorithms are thoroughly reviewed and their performance is discussed, while a broad set of examples illustrate the wide range of potential applications. The description of various numerical and analytical techniques for the computation of Lyapunov exponents offers an extensive array of tools for the characterization of phenomena such as synchronization, weak and global chaos in low and high-dimensional set-ups, and localization. This text equips readers with all the investigative expertise needed to fully explore the dynamical properties of complex systems, making it ideal for both graduate students and experienced researchers...
Scaling of Lyapunov Exponents in Homogeneous, Isotropic DNS
Fitzsimmons, Nicholas; Malaya, Nicholas; Moser, Robert
2013-11-01
Lyapunov exponents measure the rate of separation of initially infinitesimally close trajectories in a chaotic system. Using the exponents, we are able to probe the chaotic nature of homogeneous isotropic turbulence and study the instabilities of the chaotic field. The exponents are measured by calculating the instantaneous growth rate of a linear disturbance, evolved with the linearized Navier-Stokes equation, at each time step. In this talk, we examine these exponents in the context of homogeneous isotropic turbulence with two goals: 1) to investigate the scaling of the exponents with respect to the parameters of forced homogeneous isotropic turbulence, and 2) to characterize the instabilities that lead to chaos in turbulence. Specifically, we explore the scaling of the Lyapunov exponents with respect to the Reynolds number and with respect to the ratio of the integral length scale and the computational domain size.
Quantum synchronization in an optomechanical system based on Lyapunov control.
Li, Wenlin; Li, Chong; Song, Heshan
2016-06-01
We extend the concepts of quantum complete synchronization and phase synchronization, which were proposed in A. Mari et al., Phys. Rev. Lett. 111, 103605 (2013)PRLTAO0031-900710.1103/PhysRevLett.111.103605, to more widespread quantum generalized synchronization. Generalized synchronization can be considered a necessary condition or a more flexible derivative of complete synchronization, and its criterion and synchronization measure are proposed and analyzed in this paper. As examples, we consider two typical generalized synchronizations in a designed optomechanical system. Unlike the effort to construct a special coupling synchronization system, we purposefully design extra control fields based on Lyapunov control theory. We find that the Lyapunov function can adapt to more flexible control objectives, which is more suitable for generalized synchronization control, and the control fields can be achieved simply with a time-variant voltage. Finally, the existence of quantum entanglement in different generalized synchronizations is also discussed.
Geometry of dynamics, Lyapunov exponents and phase transitions
Caiani, L; Clementi, C; Pettini, M; Caiani, Lando; Casetti, Lapo; Clementi, Cecilia; Pettini, Marco
1997-01-01
The Hamiltonian dynamics of classical planar Heisenberg model is numerically investigated in two and three dimensions. By considering the dynamics as a geodesic flow on a suitable Riemannian manifold, it is possible to analytically estimate the largest Lyapunov exponent in terms of some curvature fluctuations. The agreement between numerical and analytical values for Lyapunov exponents is very good in a wide range of temperatures. Moreover, in the three dimensional case, in correspondence with the second order phase transition, the curvature fluctuations exibit a singular behaviour which is reproduced in an abstract geometric model suggesting that the phase transition might correspond to a change in the topology of the manifold whose geodesics are the motions of the system.
Practical stability of nonlinear systems
Lakshmikantham, Vangipuram; Martynyuk, Anatolii Andreevich
1990-01-01
This is the first book that deals with practical stability and its development. It presents a systematic study of the theory of practical stability in terms of two different measures and arbitrary sets and demonstrates the manifestations of general Lyapunov's method by showing how this effective technique can be adapted to investigate various apparently diverse nonlinear problems including control systems and multivalued differential equations.
Lyapunov functions for a dengue disease transmission model
Energy Technology Data Exchange (ETDEWEB)
Tewa, Jean Jules [Department of Mathematics, Faculty of Science, University of Yaounde I, P.O. Box 812, Yaounde (Cameroon)], E-mail: tewa@univ-metz.fr; Dimi, Jean Luc [Department of Mathematics, Faculty of Science, University Marien Ngouabi, P.O. Box 69, Brazzaville (Congo, The Democratic Republic of the)], E-mail: jldimi@yahoo.fr; Bowong, Samuel [Department of Mathematics and Computer Science, Faculty of Science, University of Douala, P.O. Box 24157, Douala (Cameroon)], E-mail: samuelbowong@yahoo.fr
2009-01-30
In this paper, we study a model for the dynamics of dengue fever when only one type of virus is present. For this model, Lyapunov functions are used to show that when the basic reproduction ratio is less than or equal to one, the disease-free equilibrium is globally asymptotically stable, and when it is greater than one there is an endemic equilibrium which is also globally asymptotically stable.
Analysis of Lyapunov Method for Control of Quantum States
Wang, Xiaoting; Schirmer, Sonia
2009-01-01
The natural trajectory tracking problem is studied for generic quantum states represented by density operators. A control design based on the Hilbert-Schmidt distance as a Lyapunov function is considered. The control dynamics is redefined on an extended space where the LaSalle invariance principle can be correctly applied even for non-stationary target states. LaSalle's invariance principle is used to derive a general characterization of the invariant set, which is shown to always contain the...
A local Echo State Property through the largest Lyapunov exponent.
Wainrib, Gilles; Galtier, Mathieu N
2016-04-01
Echo State Networks are efficient time-series predictors, which highly depend on the value of the spectral radius of the reservoir connectivity matrix. Based on recent results on the mean field theory of driven random recurrent neural networks, enabling the computation of the largest Lyapunov exponent of an ESN, we develop a cheap algorithm to establish a local and operational version of the Echo State Property.
The de Finetti theorem for test spaces
Barrett, Jonathan; Leifer, Matthew
2009-03-01
We prove a de Finetti theorem for exchangeable sequences of states on test spaces, where a test space is a generalization of the sample space of classical probability theory and the Hilbert space of quantum theory. The standard classical and quantum de Finetti theorems are obtained as special cases. By working in a test space framework, the common features that are responsible for the existence of these theorems are elucidated. In addition, the test space framework is general enough to imply a de Finetti theorem for classical processes. We conclude by discussing the ways in which our assumptions may fail, leading to probabilistic models that do not have a de Finetti theorem.
The de Finetti theorem for test spaces
Energy Technology Data Exchange (ETDEWEB)
Barrett, Jonathan [H H Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL (United Kingdom); Leifer, Matthew [Institute for Quantum Computing, University of Waterloo, 200 University Avenue West, Waterloo, Ontario N2L 3G1 (Canada)], E-mail: j.barrett@bristol.ac.uk, E-mail: matt@mattleifer.info
2009-03-15
We prove a de Finetti theorem for exchangeable sequences of states on test spaces, where a test space is a generalization of the sample space of classical probability theory and the Hilbert space of quantum theory. The standard classical and quantum de Finetti theorems are obtained as special cases. By working in a test space framework, the common features that are responsible for the existence of these theorems are elucidated. In addition, the test space framework is general enough to imply a de Finetti theorem for classical processes. We conclude by discussing the ways in which our assumptions may fail, leading to probabilistic models that do not have a de Finetti theorem.
Improved Stabilization Criteria for Neutral Time-Delay Systems
Lianglin Xiong; Haiyang Zhang; Yongkun Li; Zixin Liu
2016-01-01
This paper addresses the stabilization conditions for neutral systems with mixed time delays. By constructing a novel class of Lyapunov functionals which contains an augmented Lyapunov functional, using a new class of improved Jensen’s like inequalities, two improved delay-dependent stability criteria are firstly established. Next, state feedback controllers are designed according to the stability conditions in different cases. Finally, five numerical examples are provided to demonstrate the ...
Lyapunov exponents for one-dimensional aperiodic photonic bandgap structures
Kissel, Glen J.
2011-10-01
Existing in the "gray area" between perfectly periodic and purely randomized photonic bandgap structures are the socalled aperoidic structures whose layers are chosen according to some deterministic rule. We consider here a onedimensional photonic bandgap structure, a quarter-wave stack, with the layer thickness of one of the bilayers subject to being either thin or thick according to five deterministic sequence rules and binary random selection. To produce these aperiodic structures we examine the following sequences: Fibonacci, Thue-Morse, Period doubling, Rudin-Shapiro, as well as the triadic Cantor sequence. We model these structures numerically with a long chain (approximately 5,000,000) of transfer matrices, and then use the reliable algorithm of Wolf to calculate the (upper) Lyapunov exponent for the long product of matrices. The Lyapunov exponent is the statistically well-behaved variable used to characterize the Anderson localization effect (exponential confinement) when the layers are randomized, so its calculation allows us to more precisely compare the purely randomized structure with its aperiodic counterparts. It is found that the aperiodic photonic systems show much fine structure in their Lyapunov exponents as a function of frequency, and, in a number of cases, the exponents are quite obviously fractal.
Institute of Scientific and Technical Information of China (English)
Shu-jun Liu; Ji-feng Zhang; Zhong-ping Jiang
2008-01-01
In this paper, the property of practical input-to-state stability and its application to stability of cascaded nonlinear systems are investigated in the stochastic framework. Firstly, the notion of (practical)stochastic input-to-state stability with respect to a stochastic input is introduced, and then by the method of changing supply functions, (a) an (practical) SISS-Lyapunov function for the overall system is obtained from the corresponding Lyapunov functions for cascaded (practical) SISS subsystems.
Multiwavelet sampling theorem in Sobolev spaces
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
This paper is to establish the multiwavelet sampling theorem in Sobolev spaces. Sampling theorem plays a very important role in digital signal communication. The most classical sampling theorem is Shannon sampling theorem, which works for bandlimited signals. Recently, sampling theorems in wavelets or multiwavelets subspaces are extensively studied in the literature. In this paper, we firstly propose the concept of dual multiwavelet frames in dual Sobolev spaces (H s (R) , H-s (R)). Then we construct a special class of dual multiwavelet frames, from which the multiwavelet sampling theorem in Sobolev spaces is obtained. That is, for any f ∈ H s (R) with s > 1/2, it can be exactly recovered by its samples. Especially, the sampling theorem works for continuous signals in L 2 (R), whose Sobolev exponents are greater than 1 /2.
Detection of the onset of numerical chaotic instabilities by lyapunov exponents
Directory of Open Access Journals (Sweden)
Alicia Serfaty De Markus
2001-01-01
Full Text Available It is commonly found in the fixed-step numerical integration of nonlinear differential equations that the size of the integration step is opposite related to the numerical stability of the scheme and to the speed of computation. We present a procedure that establishes a criterion to select the largest possible step size before the onset of chaotic numerical instabilities, based upon the observation that computational chaos does not occur in a smooth, continuous way, but rather abruptly, as detected by examining the largest Lyapunov exponent as a function of the step size. For completeness, examination of the bifurcation diagrams with the step reveals the complexity imposed by the algorithmic discretization, showing the robustness of a scheme to numerical instabilities, illustrated here for explicit and implicit Euler schemes. An example of numerical suppression of chaos is also provided.
A unifying Lyapunov-based framework for the event-triggered control of nonlinear systems
Postoyan, Romain; Nesic, Dragan; Tabuada, Paulo
2011-01-01
We present a prescriptive framework for the event-triggered control of nonlinear systems. Rather than closing the loop periodically, as traditionally done in digital control, in event-triggered implementations the loop is closed according to a state-dependent criterion. Event-triggered control is especially well suited for embedded systems and networked control systems since it reduces the amount of resources needed for control such as communication bandwidth. By modeling the event-triggered implementations as hybrid systems, we provide Lyapunov-based conditions to guarantee the stability of the resulting closed-loop system and explain how they can be utilized to synthesize event-triggering rules. We illustrate the generality of the approach by showing how it encompasses several existing event-triggering policies and by developing new strategies which further reduce the resources needed for control.
Directory of Open Access Journals (Sweden)
Prechanon Kumkratug
2011-01-01
Full Text Available Problem statement: The disturbance in power system is unavoidable situation. It causes in power system oscillation. Approach: This study applied the Static Var Compensator (SVC to damp power system oscillation. The stability criterion of the Lyapunov is applied to derive the control strategy of SVC. The simulation results are tested on a Single Machine Infinite bus. The proposed method is equipped in sample system with disturbance. The generator rotor angle curve of the system without and with a SVC is plotted and compared for various cases. Results: It was found that the system without a SVC has high variation whereas that of the system with a SVC has much smaller variation. Conclusion: From the simulation results, the SVC can damp power system oscillaton.
Wen, Guanghui; Yu, Wenwu; Hu, Guoqiang; Cao, Jinde; Yu, Xinghuo
2015-12-01
This paper studies the global pinning synchronization problem for a class of complex networks with switching directed topologies. The common assumption in the existing related literature that each possible network topology contains a directed spanning tree is removed in this paper. Using tools from M -matrix theory and stability analysis of the switched nonlinear systems, a new kind of network topology-dependent multiple Lyapunov functions is proposed for analyzing the synchronization behavior of the whole network. It is theoretically shown that the global pinning synchronization in switched complex networks can be ensured if some nodes are appropriately pinned and the coupling is carefully selected. Interesting issues of how many and which nodes should be pinned for possibly realizing global synchronization are further addressed. Finally, some numerical simulations on coupled neural networks are provided to verify the theoretical results.
Navier Stokes Theorem in Hydrology
Narayanan, M.
2005-12-01
In a paper presented at the 2004 AGU International Conference, the author outlined and stressed the importance of studying and teaching certain important mathematical techniques while developing a course in Hydrology and Fluid Mechanics. The Navier-Stokes equations are the foundation of fluid mechanics, and Stokes' theorem is used in nearly every branch of mechanics as well as electromagnetics. Stokes' Theorem also plays a vital role in many secondary theorems such as those pertaining to vorticity and circulation. Mathematically expressed, Stokes' theorem can be expressed by considering a surface S having a bounding curve C. Here, V is any sufficiently smooth vector field defined on the surface and its bounding curve C. In an article entitled "Corrections to Fluid Dynamics" R. F. Streater, (Open Systems and Information Dynamics, 10, 3-30, 2003.) proposes a kinetic model of a fluid in which five macroscopic fields, the mass, energy, and three components of momentum, are conserved. The dynamics is constructed using the methods of statistical dynamics, and results in a non-linear discrete-time Markov chain for random fields on a lattice. In the continuum limit he obtains a non-linear coupled parabolic system of field equations, showing a correction to the Navier-Stokes equations. In 2001, David Hoff published an article in Journees Equations aux derivees partielles. (Art. No. 7, 9 p.). His paper is entitled : Dynamics of Singularity Surfaces for Compressible Navier-Stokes Flows in Two Space Dimensions. In his paper, David Hoff proves the global existence of solutions of the Navier-Stokes equations of compressible, barotropic flow in two space dimensions with piecewise smooth initial data. These solutions remain piecewise smooth for all time, retaining simple jump discontinuities in the density and in the divergence of the velocity across a smooth curve, which is convected with the flow. The strengths of these discontinuities are shown to decay exponentially in time
Expectation Value in Bell's Theorem
Wang, Zheng-Chuan
2006-01-01
We will demonstrate in this paper that Bell's theorem (Bell's inequality) does not really conflict with quantum mechanics, the controversy between them originates from the different definitions for the expectation value using the probability distribution in Bell's inequality and the expectation value in quantum mechanics. We can not use quantum mechanical expectation value measured in experiments to show the violation of Bell's inequality and then further deny the local hidden-variables theor...
ANDRAGOGY OF DEVELOPMENT: BASIC THEOREMS
Directory of Open Access Journals (Sweden)
A. G. Teslinov
2016-01-01
Full Text Available The article presents criticism of the state of scientific knowledge about adult education and provides the reasons for the choice of directions of its development.Methods. The approach to the substantiation of directions of development of andragogy includes aspectual analysis of scientific rhetoric of adult education; summarizing the symptoms and causes of the problems of educational practice examples of education managers; the analysis of the status of andragogy as a scientific paradigm; a conceptual analysis of the key theses of the modern synthesis of andragogy and the provisions for developmental adult education.Results and scientific novelty. Four theorems are formulated that specify the complete set of propositions about a developmental approach to adult education. These theorems are presented as a scientific hypothesis about the features of the approach. The theorems are proved, and the substantiation of the conditions of emergence of the adult education of educational properties is described. The idea of adult education as a developing culture is in the centre of reasoning. It is shown that the assertions of theorems form the conceptual core of the scientific branches in adult education – andragogy of development. The effect of the practical interpretation of its provisions is disclosed.Practical significance. Disclosed meanings and recommendations may be oriented to developers of educational systems and media for adults while creating the developmental components. These references will help to overcome the evident trend information of adult education to the "pulling" them up to continually outdated standards, and to give it the look of a truly developing technology.
An improvement of Papadakis' theorem
Institute of Scientific and Technical Information of China (English)
ZHANG Zhihua; MU Lehua; ZHANG Peixuan
2004-01-01
There exist many orthonormal wavelets which cannot be derived by multiresolution analysis (MRA) with a single scaling function.In 2000,Papadakis announced that any orthonormal wavelet is derived by a generalized MRA with countable scaling functions at most.We improve Papadakis' theorem and find that for any othonormal wavelet,the least number of the corresponding scaling functions is just the essential supremum of the dimension function of the orthonormal wavelet.Moreover,we construct directly the fewest scaling functions.
Pythagoras Theorem and Relativistic Kinematics
Mulaj, Zenun; Dhoqina, Polikron
2010-01-01
In two inertial frames that move in a particular direction, may be registered a light signal that propagates in an angle with this direction. Applying Pythagoras theorem and principles of STR in both systems, we can derive all relativistic kinematics relations like the relativity of simultaneity of events, of the time interval, of the length of objects, of the velocity of the material point, Lorentz transformations, Doppler effect and stellar aberration.
Compactness theorems of fuzzy semantics
Institute of Scientific and Technical Information of China (English)
无
2000-01-01
The relationship among diverse fuzzy semantics vs. the corresponding logic consequence operators has been analyzed systematically. The results that compactness and logical compactness of fuzzy semantics are equivalent to compactness and continuity of the logic consequence operator induced by the semantics respectively have been proved under certain conditions. A general compactness theorem of fuzzy semantics have been established which says that every fuzzy semantics defined on a free algebra with members corresponding to continuous functions is compact.
Software Reliability through Theorem Proving
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S.G.K. Murthy
2009-05-01
Full Text Available Improving software reliability of mission-critical systems is widely recognised as one of the major challenges. Early detection of errors in software requirements, designs and implementation, need rigorous verification and validation techniques. Several techniques comprising static and dynamic testing approaches are used to improve reliability of mission critical software; however it is hard to balance development time and budget with software reliability. Particularly using dynamic testing techniques, it is hard to ensure software reliability, as exhaustive testing is not possible. On the other hand, formal verification techniques utilise mathematical logic to prove correctness of the software based on given specifications, which in turn improves the reliability of the software. Theorem proving is a powerful formal verification technique that enhances the software reliability for missioncritical aerospace applications. This paper discusses the issues related to software reliability and theorem proving used to enhance software reliability through formal verification technique, based on the experiences with STeP tool, using the conventional and internationally accepted methodologies, models, theorem proving techniques available in the tool without proposing a new model.Defence Science Journal, 2009, 59(3, pp.314-317, DOI:http://dx.doi.org/10.14429/dsj.59.1527
Schubert, Sebastian; Lucarini, Valerio
2016-04-01
One of the most relevant weather regimes in the mid latitudes atmosphere is the persistent deviation from the approximately zonally symmetric jet stream to the emergence of so-called blocking patterns. Such configurations are usually connected to exceptional local stability properties of the flow which come along with an improved local forecast skills during the phenomenon. It is instead extremely hard to predict onset and decay of blockings. Covariant Lyapunov Vectors (CLVs) offer a suitable characterization of the linear stability of a chaotic flow, since they represent the full tangent linear dynamics by a covariant basis which explores linear perturbations at all time scales. Therefore, we will test whether CLVs feature a signature of the blockings. We examine the CLVs for a quasi-geostrophic beta-plane two-layer model in a periodic channel baroclinically driven by a meridional temperature gradient ΔT. An orographic forcing enhances the emergence of localized blocked regimes. We detect the blocking events of the channel flow with a Tibaldi-Molteni scheme adapted to the periodic channel. When blocking occurs, the global growth rates of the fastest growing CLVs are significantly higher. Hence against intuition, globally the circulation is more unstable in blocked phases. Such an increase in the finite time Lyapunov exponents with respect to the long term average is attributed to stronger barotropic and baroclinic conversion in the case of high temperature gradients, while for low values of ΔT, the effect is only due to stronger barotropic instability. For the localization of the CLVs, we compare the meridionally averaged variance of the CLVs during blocked and unblocked phases. We find that on average the variance of the CLVs is clustered around the center of blocking. These results show that the blocked flow affects all time scales and processes described by the CLVs.
Schubert, Sebastian; Lucarini, Valerio
2016-04-01
The classical approach for studying atmospheric variability is based on defining a background state and studying the linear stability of the small fluctuations around such a state. Weakly non-linear theories can be constructed using higher order expansions terms. While these methods have undoubtedly great value for elucidating the relevant physical processes, they are unable to follow the dynamics of a turbulent atmosphere. We provide a first example of extension of the classical stability analysis to a non-linearly evolving quasi-geostrophic flow. The so-called covariant Lyapunov vectors (CLVs) provide a covariant basis describing the directions of exponential expansion and decay of perturbations to the non-linear trajectory of the flow. We use such a formalism to re-examine the basic barotropic and baroclinic processes of the atmosphere with a quasi-geostrophic beta-plane two-layer model in a periodic channel driven by a forced meridional temperature gradient ΔT . We explore three settings of ΔT , representative of relatively weak turbulence, well-developed turbulence, and intermediate conditions. We construct the Lorenz energy cycle for each CLV describing the energy exchanges with the background state. A positive baroclinic conversion rate is a necessary but not sufficient condition of instability. Barotropic instability is present only for few very unstable CLVs for large values of ΔT. Slowly growing and decaying hydrodynamic Lyapunov modes closely mirror the properties of the background flow. Following classical necessary conditions for barotropic/baroclinic instability, we find a clear relationship between the properties of the eddy fluxes of a CLV and its instability. CLVs with positive baroclinic conversion seem to form a set of modes for constructing a reduced model of the atmosphere dynamics.
Stability analysis of nonlinear systems
Lakshmikantham, Vangipuram; Martynyuk, Anatoly A
2015-01-01
The book investigates stability theory in terms of two different measure, exhibiting the advantage of employing families of Lyapunov functions and treats the theory of a variety of inequalities, clearly bringing out the underlying theme. It also demonstrates manifestations of the general Lyapunov method, showing how this technique can be adapted to various apparently diverse nonlinear problems. Furthermore it discusses the application of theoretical results to several different models chosen from real world phenomena, furnishing data that is particularly relevant for practitioners. Stability Analysis of Nonlinear Systems is an invaluable single-sourse reference for industrial and applied mathematicians, statisticians, engineers, researchers in the applied sciences, and graduate students studying differential equations.
Homological stability of diffeomorphism groups
DEFF Research Database (Denmark)
Berglund, Alexander; Madsen, Ib Henning
2013-01-01
In this paper we prove a stability theorem for block diffeomorphisms of 2d -dimensional manifolds that are connected sums of S d ×S d . Combining this with a recent theorem of S. Galatius and O. Randal-Williams and Morlet’s lemma of disjunction, we determine the homology of the classifying space ...
Strict Stability of Impulsive Differential Equations
Institute of Scientific and Technical Information of China (English)
Yu ZHANG; Ji Tao SUN
2006-01-01
In this paper, we will extend the strict stability to impulsive differential equations. By using Lyapunov functions, we will get some criteria for the strict stability of impulsive differential equations, and we can see that impulses do contribute to the system's strict stability behavior. An example is also given in this paper to illustrate the efficiency of the obtained results.
Stability of Fractional Order Switching Systems
HosseinNia, S Hassan; Vinagre, Blas M
2012-01-01
This paper addresses the stabilization issue for fractional order switching systems. Common Lyapunov method is generalized for fractional order systems and frequency domain stability equivalent to this method is proposed to prove the quadratic stability. Some examples are given to show the applicability and effectiveness of the proposed theory.
Stability of stochastic switched SIRS models
Meng, Xiaoying; Liu, Xinzhi; Deng, Feiqi
2011-11-01
Stochastic stability problems of a stochastic switched SIRS model with or without distributed time delay are considered. By utilizing the Lyapunov methods, sufficient stability conditions of the disease-free equilibrium are established. Stability conditions about the subsystem of the stochastic switched SIRS systems are also obtained.
Institute of Scientific and Technical Information of China (English)
Wu Fuke; Hu Shigeng; Mao Xuerong
2011-01-01
This paper establishes the Razumikhin-type theorem on stability for neutral stochastic functional differential equations with unbounded delay.To overcome difficulties from unbounded delay,we develop several different techniques to investigate stability.To show our idea clearly,we examine neutral stochastic delay differential equations with unbounded delay and linear neutral stochastic Volterra unbounded-delay-integro-differential equations.
Directory of Open Access Journals (Sweden)
Muhammad H. Al-Malack
2016-07-01
Full Text Available Fuel oil flyash (FFA produced in power and water desalination plants firing crude oils in the Kingdom of Saudi Arabia is being disposed in landfills, which increases the burden on the environment, therefore, FFA utilization must be encouraged. In the current research, the effect of adding FFA on the engineering properties of two indigenous soils, namely sand and marl, was investigated. FFA was added at concentrations of 5%, 10% and 15% to both soils with and without the addition of Portland cement. Mixtures of the stabilized soils were thoroughly evaluated using compaction, California Bearing Ratio (CBR, unconfined compressive strength (USC and durability tests. Results of these tests indicated that stabilized sand mixtures could not attain the ACI strength requirements. However, marl was found to satisfy the ACI strength requirement when only 5% of FFA was added together with 5% of cement. When the FFA was increased to 10% and 15%, the mixture’s strength was found to decrease to values below the ACI requirements. Results of the Toxicity Characteristics Leaching Procedure (TCLP, which was performed on samples that passed the ACI requirements, indicated that FFA must be cautiously used in soil stabilization.
Directory of Open Access Journals (Sweden)
Ying Wang
2014-01-01
Full Text Available We investigate the dynamical behaviors of a class of discrete SIRS epidemic models with nonlinear incidence rate and varying population sizes. The model is required to possess different death rates for the susceptible, infectious, recovered, and constant recruitment into the susceptible class, infectious class, and recovered class, respectively. By using the inductive method, the positivity and boundedness of all solutions are obtained. Furthermore, by constructing new discrete type Lyapunov functions, the sufficient and necessary conditions on the global asymptotic stability of the disease-free equilibrium and endemic equilibrium are established.
Stability and Speed Control of a Series-Wound DC Motor
Khoroshun, A. S.
2016-07-01
A speed control for a series-wound DC motor is proposed. It is shown that steady-state rotation is stabile and robust. Stability is analyzed using a quadratic Lyapunov function. Its explicit expression is derived
The classical version of Stokes' Theorem revisited
DEFF Research Database (Denmark)
Markvorsen, Steen
2008-01-01
Using only fairly simple and elementary considerations - essentially from first year undergraduate mathematics - we show how the classical Stokes' theorem for any given surface and vector field in $\\mathbb{R}^{3}$ follows from an application of Gauss' divergence theorem to a suitable modification...... of the vector field in a tubular shell around the given surface. The two stated classical theorems are (like the fundamental theorem of calculus) nothing but shadows of the general version of Stokes' theorem for differential forms on manifolds. The main points in the present paper, however, is firstly...... exercise, which simply relates the concepts of divergence and curl on the local differential level. The rest of the paper uses only integration in $1$, $2$, and $3$ variables together with a 'fattening' technique for surfaces and the inverse function theorem....
On Krasnoselskii's Cone Fixed Point Theorem
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Man Kam Kwong
2008-04-01
Full Text Available In recent years, the Krasnoselskii fixed point theorem for cone maps and its many generalizations have been successfully applied to establish the existence of multiple solutions in the study of boundary value problems of various types. In the first part of this paper, we revisit the Krasnoselskii theorem, in a more topological perspective, and show that it can be deduced in an elementary way from the classical Brouwer-Schauder theorem. This viewpoint also leads to a topology-theoretic generalization of the theorem. In the second part of the paper, we extend the cone theorem in a different direction using the notion of retraction and show that a stronger form of the often cited Leggett-Williams theorem is a special case of this extension.
The Helmholtz theorem and retarded fields
Heras, Ricardo
2016-11-01
Textbooks frequently use the Helmholtz theorem to derive expressions for electrostatic and magnetostatic fields but they do not usually apply this theorem to derive expressions for time-dependent electric and magnetic fields, even when there is no formal objection to doing so because the proof of the theorem does not involve time derivatives but only spatial derivatives. Here we address the question as to whether the Helmholtz theorem is useful in deriving expressions for the fields of Maxwell’s equations. We show that when this theorem is applied to Maxwell’s equations we obtain instantaneous expressions of the electric and magnetic fields, which are formally correct but of little practical usefulness. We then discuss two generalizations of the theorem which are shown to be useful in deriving the retarded fields.
The Helmholtz theorem and retarded fields
Heras, Ricardo
2016-01-01
Textbooks frequently use the Helmholtz theorem to derive expressions for the electrostatic and magnetostatic fields but they do not usually apply this theorem to derive expressions for the time-dependent electric and magnetic fields, even when there is no formal objection to doing so because the proof of the theorem does not involve time derivatives but only spatial derivatives. Here we address the question as to whether the Helmholtz theorem is useful to derive expressions for the fields of Maxwell's equations. We show that when this theorem is applied to Maxwell's equations we obtain instantaneous expressions of the electric and magnetic fields, which are formally correct but of little practical usefulness. We then discuss two generalizations of the theorem which are shown to be useful to derive the retarded fields.
Uniformly Asymptotic Stability and Existence of Periodic Solutions and Almost Periodic Solutions
Institute of Scientific and Technical Information of China (English)
林发兴
1994-01-01
We introduce the concepts of uniformly stable solutions and uniformly-asymptotically stable solutions, and then give the relation between Lyapunov function and those concepts. Furthermore, we establish the theorem that an almost periodic system or periodic system has uniformly-asymptotically stable solutions and Lipschitz’ condition has an almost periodic solution or periodic solution.
Point Stabilization for an Underactuated AUV in the Presence of Ocean Currents
Directory of Open Access Journals (Sweden)
Zaopeng Dong
2015-07-01
Full Text Available This paper presents a state-feedback-based backstepping control algorithm to address the point stabilization (or set-point regulation control problem for an underactuated autonomous underwater vehicle (AUV in the presence of constant and irrotational ocean current disturbance. A nonlinear three degree of freedom dynamic model in the horizontal plane for an AUV without symmetry fore/aft is considered. The expression of the relationship between the desired heading angle of the AUV and direction angle of the ocean current, which is a necessary condition for precise point stabilization control of an underactuated AUV in the presence of ocean current disturbance is firstly discussed in this paper. The proposed backstepping control law for point stabilization has further been enriched by incorporating an additional integral action for enhancing the steady state performance of the AUV control system, while practical asymptotic stability analysis of the system is carried out using Lyapunov theory and Barbalat's Lemma. Simulation experiments of an underactuated AUV verify the theorem proposed and demonstrate the effectiveness of the controller.
Point Stabilization for an Underactuated AUV in the Presence of Ocean Currents
Directory of Open Access Journals (Sweden)
Zaopeng Dong
2015-07-01
Full Text Available This paper presents a state-feedback-based backstepping control algorithm to address the point stabilization (or set-point regulation control problem for an underactuated autonomous underwater vehicle (AUV in the presence of constant and irrotational ocean current disturbance. A nonlinear three degree of freedom dynamic model in the horizontal plane for an AUV without symmetry fore/aft is considered. The expression of the relationship between the desired heading angle of the AUV and direction angle of the ocean current, which is a necessary condition for precise point stabilization control of an underactuated AUV in the presence of ocean current disturbance is firstly discussed in this paper. The proposed backstepping control law for point stabilization has further been enriched by incorporating an additional integral action for enhancing the steady state performance of the AUV control system, while practical asymptotic stability analysis of the system is carried out using Lyapunov theory and Barbalat’s Lemma. Simulation experiments of an underactuated AUV verify the theorem proposed and demonstrate the effectiveness of the controller.
Continuation of probability density functions using a generalized Lyapunov approach
Energy Technology Data Exchange (ETDEWEB)
Baars, S., E-mail: s.baars@rug.nl [Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, P.O. Box 407, 9700 AK Groningen (Netherlands); Viebahn, J.P., E-mail: viebahn@cwi.nl [Centrum Wiskunde & Informatica (CWI), P.O. Box 94079, 1090 GB, Amsterdam (Netherlands); Mulder, T.E., E-mail: t.e.mulder@uu.nl [Institute for Marine and Atmospheric research Utrecht, Department of Physics and Astronomy, Utrecht University, Princetonplein 5, 3584 CC Utrecht (Netherlands); Kuehn, C., E-mail: ckuehn@ma.tum.de [Technical University of Munich, Faculty of Mathematics, Boltzmannstr. 3, 85748 Garching bei München (Germany); Wubs, F.W., E-mail: f.w.wubs@rug.nl [Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, P.O. Box 407, 9700 AK Groningen (Netherlands); Dijkstra, H.A., E-mail: h.a.dijkstra@uu.nl [Institute for Marine and Atmospheric research Utrecht, Department of Physics and Astronomy, Utrecht University, Princetonplein 5, 3584 CC Utrecht (Netherlands); School of Chemical and Biomolecular Engineering, Cornell University, Ithaca, NY (United States)
2017-05-01
Techniques from numerical bifurcation theory are very useful to study transitions between steady fluid flow patterns and the instabilities involved. Here, we provide computational methodology to use parameter continuation in determining probability density functions of systems of stochastic partial differential equations near fixed points, under a small noise approximation. Key innovation is the efficient solution of a generalized Lyapunov equation using an iterative method involving low-rank approximations. We apply and illustrate the capabilities of the method using a problem in physical oceanography, i.e. the occurrence of multiple steady states of the Atlantic Ocean circulation.
Abstraction of Continuous Dynamical Systems Utilizing Lyapunov Functions
DEFF Research Database (Denmark)
Sloth, Christoffer; Wisniewski, Rafal
2010-01-01
This paper considers the development of a method for abstracting continuous dynamical systems by timed automata. The method is based on partitioning the state space of dynamical systems with invariant sets, which form cells representing locations of the timed automata. To enable verification...... of the dynamical system based on the abstraction, conditions for obtaining sound, complete, and refinable abstractions are set up. It is proposed to partition the state space utilizing sub-level sets of Lyapunov functions, since they are positive invariant sets. The existence of sound abstractions for Morse......-Smale systems and complete and refinable abstractions for linear systems are shown....
Suppressing chaos via Lyapunov-Krasovskii's method
Energy Technology Data Exchange (ETDEWEB)
Kuang, J.L. [Faculty of Science and Engineering, City University of Hong Kong, Hong Kong (China)] e-mail: kuangjinlu@hotmail.com; Meehan, P.A. [Department of Mechanical Engineering, University of Queensland, Brisbane, Qld 4072 (Australia)] e-mail: meehan@uq.edu.au; Leung, A.Y.T. [Faculty of Science and Engineering, City University of Hong Kong, Hong Kong (China)] e-mail: bcaleung@cityu.edu.hk
2006-03-01
An algorithm for suppressing the chaotic oscillations in non-linear dynamical systems with singular Jacobian matrices is developed using a linear feedback control law based upon the Lyapunov-Krasovskii (LK) method. It appears that the LK method can serve effectively as a generalised method for the suppression of chaotic oscillations for a wide range of systems. Based on this method, the resulting conditions for undisturbed motions to be locally or globally stable are sufficient and conservative. The generalized Lorenz system and disturbed gyrostat equations are exemplified for the validation of the proposed feedback control rule.
Symbolic logic and mechanical theorem proving
Chang, Chin-Liang
1969-01-01
This book contains an introduction to symbolic logic and a thorough discussion of mechanical theorem proving and its applications. The book consists of three major parts. Chapters 2 and 3 constitute an introduction to symbolic logic. Chapters 4-9 introduce several techniques in mechanical theorem proving, and Chapters 10 an 11 show how theorem proving can be applied to various areas such as question answering, problem solving, program analysis, and program synthesis.
Expanding the Interaction Equivalency Theorem
Directory of Open Access Journals (Sweden)
Brenda Cecilia Padilla Rodriguez
2015-06-01
Full Text Available Although interaction is recognised as a key element for learning, its incorporation in online courses can be challenging. The interaction equivalency theorem provides guidelines: Meaningful learning can be supported as long as one of three types of interactions (learner-content, learner-teacher and learner-learner is present at a high level. This study sought to apply this theorem to the corporate sector, and to expand it to include other indicators of course effectiveness: satisfaction, knowledge transfer, business results and return on expectations. A large Mexican organisation participated in this research, with 146 learners, 30 teachers and 3 academic assistants. Three versions of an online course were designed, each emphasising a different type of interaction. Data were collected through surveys, exams, observations, activity logs, think aloud protocols and sales records. All course versions yielded high levels of effectiveness, in terms of satisfaction, learning and return on expectations. Yet, course design did not dictate the types of interactions in which students engaged within the courses. Findings suggest that the interaction equivalency theorem can be reformulated as follows: In corporate settings, an online course can be effective in terms of satisfaction, learning, knowledge transfer, business results and return on expectations, as long as (a at least one of three types of interaction (learner-content, learner-teacher or learner-learner features prominently in the design of the course, and (b course delivery is consistent with the chosen type of interaction. Focusing on only one type of interaction carries a high risk of confusion, disengagement or missed learning opportunities, which can be managed by incorporating other forms of interactions.
Construction of a Smooth Lyapunov Function for the Robust and Exact Second-Order Differentiator
2016-01-01
Differentiators play an important role in (continuous) feedback control systems. In particular, the robust and exact second-order differentiator has shown some very interesting properties and it has been used successfully in sliding mode control, in spite of the lack of a Lyapunov based procedure to design its gains. As contribution of this paper, we provide a constructive method to determine a differentiable Lyapunov function for such a differentiator. Moreover, the Lyapunov function is used...
Covariant hydrodynamic Lyapunov modes and strong stochasticity threshold in Hamiltonian lattices.
Romero-Bastida, M; Pazó, Diego; López, Juan M
2012-02-01
We scrutinize the reliability of covariant and Gram-Schmidt Lyapunov vectors for capturing hydrodynamic Lyapunov modes (HLMs) in one-dimensional Hamiltonian lattices. We show that, in contrast with previous claims, HLMs do exist for any energy density, so that strong chaos is not essential for the appearance of genuine (covariant) HLMs. In contrast, Gram-Schmidt Lyapunov vectors lead to misleading results concerning the existence of HLMs in the case of weak chaos.
Inverting the central limit theorem
Navascues, Miguel; Villanueva, Ignacio
2011-01-01
The central limit theorem states that the sum of N independently distributed n-tuples of real variables (subject to appropriate normalization) tends to a multivariate gaussian distribution for large N. Here we propose to invert this argument: given a set of n correlated gaussian variables, we try to infer information about the structure of the discrete microscopic probability distributions whose convolution generated such a macroscopic behavior. The techniques developed along the article are applied to prove that the classical description of certain macroscopic optical experiments is infinitely more complex than the quantum one.
Comparison theorems in Riemannian geometry
Cheeger, Jeff
2008-01-01
The central theme of this book is the interaction between the curvature of a complete Riemannian manifold and its topology and global geometry. The first five chapters are preparatory in nature. They begin with a very concise introduction to Riemannian geometry, followed by an exposition of Toponogov's theorem-the first such treatment in a book in English. Next comes a detailed presentation of homogeneous spaces in which the main goal is to find formulas for their curvature. A quick chapter of Morse theory is followed by one on the injectivity radius. Chapters 6-9 deal with many of the most re
Computational Stability Analysis of Lotka-Volterra Systems
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Polcz Péter
2016-12-01
Full Text Available This paper concerns the computational stability analysis of locally stable Lotka-Volterra (LV systems by searching for appropriate Lyapunov functions in a general quadratic form composed of higher order monomial terms. The Lyapunov conditions are ensured through the solution of linear matrix inequalities. The stability region is estimated by determining the level set of the Lyapunov function within a suitable convex domain. The paper includes interesting computational results and discussion on the stability regions of higher (3,4 dimensional LV models as well as on the monomial selection for constructing the Lyapunov functions. Finally, the stability region is estimated of an uncertain 2D LV system with an uncertain interior locally stable equilibrium point.
ORIGINAL ARTICLE Stability Analysis of Delayed Cournot Model in ...
African Journals Online (AJOL)
HP
and Lyapunov method of nonlinear stability analysis are employed. It is ascertained ... MATLAB2012a is used to demonstrate the applicability and accuracy of the results. ...... computation, 149(3), 843-860. ... Science and Complexity, Elsevier.
A parametrization of the abstract Ramsey theorem
Mijares, Jose G
2010-01-01
We give a parametrization with perfect subsets of $2^{\\infty}$ of the abstract Ramsey theorem (see \\cite{todo}) Our main tool is an extension of the parametrized version of the combinatorial forcing developed in \\cite{nash} and \\cite{todo}, used in \\cite{mij} to the obtain a parametrization of the abstract Ellentuck theorem. As one of the consequences, we obtain a parametrized version of the Hales-Jewett theorem. Finally, we conclude that the family of perfectly ${\\cal S}$-Ramsey subsets of $2^{\\infty}\\times {\\cal R}$ is closed under the Souslin operation. {\\bf Key words and phrases}: Ramsey theorem, Ramsey space, parametrization.
A generalised Sylvester-Gallai Theorem
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L. M. Pretorius
2007-09-01
Full Text Available We give an algorithmic proof for the contrapositive of the following theorem that has recently been proved by the authors:Let S be a ﬁnite set of points in the plane, with each point coloured red, blue or with both colours. Suppose that for any two distinct points A and B in S sharing a colour k, there is a third point in S which has (inter alia the colour different from k and is collinear with A and B. Then all the points in S are collinear.This theorem is a generalization of both the Sylvester-Gallai Theorem and the Motzkin-Rabin Theorem.
A History of the Central Limit Theorem
Fischer, Hans
2011-01-01
This study discusses the history of the central limit theorem and related probabilistic limit theorems from about 1810 through 1950. In this context the book also describes the historical development of analytical probability theory and its tools, such as characteristic functions or moments. The central limit theorem was originally deduced by Laplace as a statement about approximations for the distributions of sums of independent random variables within the framework of classical probability, which focused upon specific problems and applications. Making this theorem an autonomous mathematical
A generalized preimage theorem in global analysis
Institute of Scientific and Technical Information of China (English)
MA; Jipu
2001-01-01
［1］Ma Jipu, (1.2) inverses of operators between Banach spaces and conjugacy theorem, Chinese Annals of Math., B, 1999, 20(1): 57.［2］Ma Jipu, Rank theorem of operators between Banach spaces, Science in China, Ser. A, 2000, 43(1): 1.［3］Ma Jipu, Local conjugacy theorem, rank theorems in advenced calculus and a generalized principle constructing Banach manifolds, Science in China, Ser. A, 2000, 43(12): 1233.［4］Zeidler, A. E., Nonlinear Function Analysis and Its Applications, IV: Applications to Mathematical Physics, New York: Springer-Verlag, 1988.
Directory of Open Access Journals (Sweden)
Sol Swords
2011-10-01
Full Text Available Interactive theorem proving requires a lot of human guidance. Proving a property involves (1 figuring out why it holds, then (2 coaxing the theorem prover into believing it. Both steps can take a long time. We explain how to use GL, a framework for proving finite ACL2 theorems with BDD- or SAT-based reasoning. This approach makes it unnecessary to deeply understand why a property is true, and automates the process of admitting it as a theorem. We use GL at Centaur Technology to verify execution units for x86 integer, MMX, SSE, and floating-point arithmetic.
Local Lyapunov exponents sublimiting growth rates of linear random differential equations
Siegert, Wolfgang
2009-01-01
Establishing a new concept of local Lyapunov exponents the author brings together two separate theories, namely Lyapunov exponents and the theory of large deviations. Specifically, a linear differential system is considered which is controlled by a stochastic process that during a suitable noise-intensity-dependent time is trapped near one of its so-called metastable states. The local Lyapunov exponent is then introduced as the exponential growth rate of the linear system on this time scale. Unlike classical Lyapunov exponents, which involve a limit as time increases to infinity in a fixed system, here the system itself changes as the noise intensity converges, too.
A Simple Method for the Computation of the COnditional Lyapunov Exponents
Institute of Scientific and Technical Information of China (English)
无
1999-01-01
An handy method of calculating the conditional Lyapunov exponents is put forward.Lyapunov exponents of differential dynamical systems and the conditional Lyapunov exponents can be acquired easily with the method.The method has been successfully used in kinds of synchronization ,such as continuous driving synchronization,impulsive(sporadic)driving synchronization,intermittently driving synchronization.The conditional Lyapunov exponents obtained with our method can give the largest and the best time interval for impulsive synchronization that can hardly be settled in other ways.
The matching theorems and coincidence theorems for generalized R-KKM mapping in topological spaces
Huang, Jianhua
2005-12-01
In this paper we present some new matching theorems with open cover and closed cover by using the generalized R-KKM theorems [L. Deng, X. Xia, Generalized R-KKM theorem in topological space and their applications, J. Math. Anal. Appl. 285 (2003) 679-690] in the topological spaces with property (H). As applications, some coincidence theorems are established in topological spaces. Our results extend and generalize some known results.
Institute of Scientific and Technical Information of China (English)
Ligang WU; Changhong WANG; Huijun GAO; Qingshuang ZENG
2006-01-01
The robust stability and robust sliding mode control problems are studied for a class of linear distributed time-delay systems with polytopic-type uncertainties by applying the parameter-dependent Lyapunov functional approach combining with a new method of introducing some relaxation matrices and tuning parameters, which can be chosen properly to lead to a less conservative result. First, a sufficient condition is proposed for robust stability of the autonomic system;next, the sufficient conditions of the robust stabilization controller and the existence condition of sliding mode are developed. The results are given in terms of linear matrix inequalities (LMIs), which can be solved via efficient interior-point algorithms. A numerical example is presented to illustrate the feasibility and advantages of the proposed design scheme.
An elementary derivation of the quantum virial theorem from Hellmann-Feynman theorem
İpekoğlu, Y.; Turgut, S.
2016-07-01
A simple proof of the quantum virial theorem that can be used in undergraduate courses is given. The proof proceeds by first showing that the energy eigenvalues of a Hamiltonian remain invariant under a scale transformation. Then invoking the Hellmann-Feynman theorem produces the final statement of the virial theorem.
Vallès, Jean
2012-01-01
Our aim is to prove a Poncelet type theorem for a line configuration on the complex projective. More precisely, we say that a polygon with 2n sides joining 2n vertices A1, A2,..., A2n is well inscribed in a configuration Ln of n lines if each line of the configuration contains exactly two points among A1, A2, ..., A2n. Then we prove : "Let Ln be a configuration of n lines and D a smooth conic in the complex projective plane. If it exists one polygon with 2n sides well inscribed in Ln and circumscribed around D then there are infinitely many such polygons. In particular a general point in Ln is a vertex of such a polygon." We propose an elementary proof based on Fr\\'egier's involution. We begin by recalling some facts about these involutions. Then we explore the following question : When does the product of involutions correspond to an involution? It leads to Pascal theorem, to its dual version proved by Brianchon, and to its generalization proved by M\\"obius.
Asymptotical Stability of Nonlinear Fractional Differential System with Caputo Derivative
2011-01-01
This paper deals with the stability of nonlinear fractional differential systems equipped with the Caputo derivative. At first, a sufficient condition on asymptotical stability is established by using a Lyapunov-like function. Then, the fractional differential inequalities and comparison method are applied to the analysis of the stability of fractional differential systems. In addition, some other sufficient conditions on stability are also presented.
Stability of melt crystal growth under microgravity conditions
Tatarchenko, V. A.
The conception of dynamic stability of melt crystal growth has been developed. The method based on the Lyapunov stability theory has been used to the study stability of crystallization by capillary shaping techniques including Czokhralsky, Stepanov, Kiropoulos, Verneuil and floating zone methods. Preliminary results of the stability analysis of crystallization by floating zone technique under microgravity conditions are presented here.
Directory of Open Access Journals (Sweden)
Nada S. Abdelwahab
2017-05-01
Full Text Available The present work concerns with the development of stability indicating the RP-HPLC method for simultaneous determination of guaifenesin (GUF and pseudoephedrine hydrochloride (PSH in the presence of guaifenesin related substance (Guaiacol. GUC, and in the presence of syrup excepients with minimum sample pre-treatment. In the developed RP-HPLC method efficient chromatographic separation was achieved for GUF, PSH, GUC and syrup excepients using ODS column as a stationary phase and methanol: water (50:50, v/v, pH = 4 with orthophosphoric acid as a mobile phase with a flow rate of 1 mL min−1 and UV detection at 210 nm. The chromatographic run time was approximately 10 min. Calibration curves were drawn relating the integrated area under peak to the corresponding concentrations of PSH, GUF and GUC in the range of 1–8, 1–20, 0.4–8 μg mL−1, respectively. The developed method has been validated and met the requirements delineated by ICH guidelines with respect to linearity, accuracy, precision, specificity and robustness. The validated method was successfully applied for determination of the studied drugs in triaminic chest congestion® syrup; moreover its results were statistically compared with those obtained by the official method and no significant difference was found between them.
Lyapunov-Based Feedback Preparation of GHZ Entanglement of N-Qubit Systems.
Liu, Yanan; Kuang, Sen; Cong, Shuang
2016-07-09
The Greenberger-Horne-Zeilinger (GHZ) entangled states are a typical class of entangled states in multiparticle systems and play an important role in the applications of quantum communication and quantum computation. For a general quantum system of N qubits, degenerate measurement operators are often met, which cause the convergence obstacle in the state preparation or stabilization problem. This paper first generalizes the traditional quantum state continuous reduction theory to the case of a degenerate measurement operator and chooses a measurement operator for an arbitrarily given target GHZ entangled state, then presents a state stabilization control strategy based on the Lyapunov method and achieves the feedback preparation of the target GHZ state. In our stabilization strategy, we separate the target GHZ state and all the other GHZ states that often form the equilibrium points of the closed-loop system by dividing the state space into several different regions; and formally design a switching control law between the regions, which contains the control Hamiltonians to be constructed. By analyzing the stability of the closed-loop system in the different regions, we propose a systematic method for constructing the control Hamiltonians and solve the convergence problem caused by the degenerate measurement operator. The global stability of the whole closed-loop stochastic system is strictly proved. Also, we perform some simulation experiments on a three-qubit system and prepare a three-qubit GHZ entangled state. At the same time, the simulation results show the effectiveness of the switching control law and the construction method for the control Hamiltonians proposed in this paper.
具有有界控制输入的线性时变系统闭环稳定性%Closed-loop stability of linear time-variant systems with bounded control input
Institute of Scientific and Technical Information of China (English)
张桂香
2000-01-01
In this paper,on the basis of the Lyapunov theorem,some criteria to judge the closed-loop asymptotic stability of state feedback control systems with bounded control input for linear time-variant system is proposed.The secriteria ar esimple and less conservative.%对于线性时变系统，本文基于李雅普诺夫定理，给出了具有有界控制输入的状态反馈控制系统的闭环渐近稳定性判据。这些判据具有简单的形式和较少的保守性。
Bifurcation and stability for a nonlinear parabolic partial differential equation
Chafee, N.
1973-01-01
Theorems are developed to support bifurcation and stability of nonlinear parabolic partial differential equations in the solution of the asymptotic behavior of functions with certain specified properties.