Lyapunov Functions in Dimension Estimates of Attractors of Dynamical Systems

We discuss relations between the Lyapunov dimension and the topological, Hausdorff, and fractal dimensions of attractors. For the Henon and Lorenz systems, explicit formulas characterizing the Lyapunov dimension of their attractors are obtained. Bibliography: 23 titles.     

Lyapunov Functions for Infinite-Dimensional Systems

We study Lyapunov functions for infinite-dimensional dynamical systems governed by general maximal monotone operators. We obtain a characterization of Lyapunov pairs by means of the associated Hamilton-Jacobi partial differential equations, whose solutions are meant in the viscosity sense, as evolved in works of Tataru and Crandall-Lions. Our approach also leads to a new sufficient condition for Lyapunov pairs, generalizing a classical result of Pazy.     

Smooth Lyapunov Functions for Discontinuous Stable Systems

It has been proved that a differential system d x / d t = f(t, x) with a discontinuous right-hand side admits some continuous weak Lyapunov function if and only if it is robustly stable. This paper focuses on the smoothness of such a Lyapunov function. An example of an (asymptotically) stable system for which there does not exist any (even weak) Lyapunov functions of class C ¹ is given. In the more general context of differential inclusions, the existence of a weak Lyapunov function of class C ¹ (or C ) is shown to be equivalent to the robust stability of some perturbed system obtained in introducing measurement error with respect to x and t. This condition is proved to be satisfied by most of the robustly stable systems encountered in the literature. Analogous results are given for the Lagrange stability. As an application to the study of the links between internal and external stability for control systems, an extension of a result by Bacciotti and Beccari is obtained by means of a smooth Lyapunov function associated with a robustly Lagrange stable system.     

关于切换正线性系统的共同Lyapunov函数

研究切换正线性系统的共同Lyapunov函数.分别就系统矩阵是三角阵和分块三角阵的情况给出了存在一类共同Lyapunov函数的证明,并基于证明过程给出了共同Lyapunov函数的一种求法.最后通过数值算例验证结果.     

Spectral Decompositions for the Solutions of Lyapunov Equations for Bilinear Dynamical Systems

Doklady Mathematics - In this paper, novel spectral decompositions are obtained for the solutions of generalized Lyapunov equations, which are observed in the study of controllability and...     

A New Class of Lyapunov Functions for Stability Analysis of Singular Dynamical Systems. Elements of p -Regularity Theory

Doklady Mathematics - A new approach is proposed for studying the stability of dynamical systems in the case when traditional Lyapunov functions are ineffective or not applicable for research at...     

An SOS-QE Approach to Nonlinear Gain Analysis for Polynomial Dynamical Systems

Combined use of the method of sum of squares (SOS) and quantifier elimination (QE) is discussed with regard to the problem of nonlinear gain analysis for a class of dynamical systems. SOS, a numerical method, is used to search for the structure of a gain function quickly, and QE, a symbolic method, determines all gain functions in the structure to find the minimum gain function. QE can also be used for the infeasibility check of gain structures. A proposed analysis procedure for polynomial dynamical systems prevents unnecessary searching for a gain structure. Two illustrative examples show the effectiveness of the proposed gain analysis procedure.     

A Synthesis of the Control of Dynamical Systems on the Basis of the Method of Lyapunov Functions

We consider the problem of the synthesis of a bounded control reducing a dynamical system to the given terminal state in a finite time. Two approaches to solve the problem, based on methods of the theory of stability of motion, are provided. One of them is applicable to nonlinear Lagrange mechanical systems with undetermined parameters, while another is applicable to linear systems. The characteristic property is that the Lyapunov functions are defined implicitly in both cases. We make a comparison between these approaches.     

Relaxed Quasimonotone Operators and Relaxed Quasiconvex Functions

In this paper, we introduce the class of multivalued relaxed μ quasimonotone operators and establish the existence of solutions of variational inequalities for such operators. This result is compared with a recent result of Bai et al. on densely relaxed pseudomonotone operators. A similar comparison regarding an existence result of Luc on densely pseudomonotone operators is provided. Also, we introduce a broad class of functions, called relaxed quasiconvex functions, and show that they are characterized by the relaxed μ quasimonotonicity of their subdifferentials. The results strengthen a variety of other results in the literature. This work is supported by NNSF of China (10571046) and by the GSRT of Greece (06FR-062).     

Polynomial contractions and Lyapunov regularity

The purpose of this note is twofold: to introduce the notion of polynomial contraction for a linear nonautonomous dynamics with discrete time, and to show that it persists under sufficiently small linear and nonlinear perturbations. The notion of polynomial contraction mimics the notion of exponential contraction, but with the exponential decay replaced by a polynomial decay. We show that this behavior is exhibited by a large class of dynamics, by giving necessary conditions in terms of “polynomial” Lyapunov exponents. Finally, we establish the persistence of the asymptotic stability of a polynomial contraction under sufficiently small linear and nonlinear perturbations. We also consider the case of nonuniform polynomial contractions, for which the Lyapunov stability is not uniform.     

Gap Functions and Lyapunov Functions

Equilibrium problems play a central role in the study of complex and competitive systems. Many variational formulations of these problems have been presented in these years. So, variational inequalities are very useful tools for the study of equilibrium solutions and their stability. More recently a dynamical model of equilibrium problems based on projection operators was proposed. It is designated as globally projected dynamical system (GPDS). The equilibrium points of this system are the solutions to the associated variational inequality (VI) problem. A very popular approach for finding solution of these VI and for studying its stability consists in introducing the so-called "gap-functions", while stability analysis of an equilibrium point of dynamical systems can be made by means of Lyapunov functions. In this paper we show strict relationships between gap functions and Lyapunov functions.     

On the Existence and Construction of Common Lyapunov Functions for Switched Discrete Systems

Under consideration is the problem of stability of switched discrete systems with the generalized homogeneous right-hand sides. The conditions are obtained for the existence of the common Lyapunov function, and a method for its construction is proposed in the form of a combination of the partial Lyapunov functions obtained for isolated subsystems. For a special case of linear three-dimensional systems, some algorithms are proposed for constructing common Lyapunov functions as quadratic and fourth degree forms. Some examples illustrate the effectiveness of the proposed approach.     

Lyapunov Functions for General Nonuniform Dichotomies

For nonautonomous linear equations x′ = A(t)x, we give a complete characterization of the existence of exponential behavior in terms of Lyapunov functions. In particular, we obtain an inverse theorem giving explicitly Lyapunov functions for each exponential dichotomy. The main novelty of our work is that we consider a very general type of nonuniform exponential dichotomy. This includes for example uniform exponential dichotomies, nonuniform exponential dichotomies and polynomial dichotomies. We also consider the case of different growth rates for the uniform and the nonuniform parts of the dichotomy. As an application of our work, we establish in a very direct manner the robustness of nonuniform exponential dichotomies under sufficiently small linear perturbations.     

A Bounded Stabilizing Control for Nonlinear Systems Synthesized Using Parametric Families of Lyapunov Functions

The article considers stabilization of the equilibrium point in nonlinear systems by bounded control. Stabilizing feedback is obtained by using control-Lyapunov functions (CLF) and control rescaling. CLF are applied to construct a one-parameter family of stabilizing controls, and control boundedness is achieved by specifying the law of parameter of variation (control rescaling). This method produces a bounded stabilizing feedback for a scalar-control system of a special kind.     

General Orthogonal State Variables and Applications to Lyapunov Functions and Feedback Control Systems

A general set of orthogonal state variables is formulated forlinear time-invariant systems both in the time and the frequencydomains. A general control variable u(t) and general initialconditions v(0) are considered. The meaning of orthogonalityin frequency domain is demonstrated. The applications of generalorthogonal state variables to the evaluation of Lyapunov functionsat t =0 and the evaluation of a quadratic performance criterionof feedback control systems are discussed.