Robustness via Lyapunov functions

For nonautonomous linear equations x^′=A(t)x, we give a complete characterization of nonuniform exponential dichotomies in terms of strict quadratic Lyapunov functions. Nonuniform exponential dichotomies include as a very special case uniform exponential dichotomies. In particular, we construct explicitly strict Lyapunov functions for each exponential dichotomy. As a nontrivial application, we establish in a simple and direct manner the robustness of nonuniform exponential dichotomies under sufficiently small linear perturbations. This represents a considerable simplification of former work.     

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.     

Polytope Lyapunov Functions for Stable and for Stabilizable LSS

We present a new approach for constructing polytope Lyapunov functions for continuous-time linear switching systems (LSS). This allows us to decide the stability of LSS and to compute the Lyapunov exponent with a good precision in relatively high dimensions. The same technique is also extended for stabilizability of positive systems by evaluating a polytope concave Lyapunov function (“antinorm”) in the cone. The method is based on a suitable discretization of the underlying continuous system and provides both a lower and an upper bound for the Lyapunov exponent. The absolute error in the Lyapunov exponent computation is estimated from above and proved to be linear in the dwell time. The practical efficiency of the new method is demonstrated in several examples and in the list of numerical experiments with randomly generated matrices of dimensions up to 10 (for general linear systems) and up to 100 (for positive systems). The development of the method is based on several theoretical results proved in the paper: the existence of monotone invariant norms and antinorms for positively irreducible systems, the equivalence of all contractive norms for stable systems and the linear convergence theorem.     

Nonuniform Dichotomies via Lyapunov Functions

For a nonautonomous linear equation x′ =  A(t)x we show how to characterize a nonuniform exponential dichotomy using strict Lyapunov functions. In particular, the stable and unstable subspaces are obtained from invariant families of cones determined by each Lyapunov function. We also obtain converse theorems, constructing explicitly a family of strict Lyapunov functions for each nonuniform exponential dichotomy. We emphasize that nonuniform exponential dichotomies include as a very particular case (uniform) exponential dichotomies.     

Stochastic Lyapunov Functions for a System of Nonlinear Difference Equations

We study problems related to the stability of solutions of nonlinear difference equations with random perturbations of semi-Markov type. We construct Lyapunov functions for different classes of nonlinear difference equations with semi-Markov right-hand side and establish conditions for their existence.     

滞后型脉冲泛函微分方程有界性的Lyapunov逆定理

本文通过建立滞后型脉冲泛函微分方程饱和解的存在唯一性定理,在广义常微分方程与滞后型脉冲泛函微分方程等价的基础上,研究了滞后型脉冲泛函微分方程关于一致有界性的Lyapunov逆定理.     

Impulsive stabilization of delay differential systems via the Lyapunov–Razumikhin method

This work studies global exponential stability of impulsive delay differential systems. By employing the Razumikhin technique and Lyapunov functions, several global exponential stability criteria are established for general impulsive delay differential equations. Our results show that delay differential equations may be exponentially stabilized by impulses. An example and its simulation are also given to illustrate our results.     

Solvability for Impulsive Neutral Integro-Differential Equations with State-Dependent Delay via Fractional Operators

This paper is mainly concerned with the existence of mild solutions for a first-order impulsive neutral integro-differential equation with state-dependent delay. We assume that the undelayed part generates an analytic resolvent operator and transforms it into an integral equation. By using a fixed-point theorem for condensing maps combined with theories of analytic resolvent operators, we prove some existence theorems. As an application of these main theorems, some practical consequences are derived.     

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.     

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.     

Quadratic Lyapunov functions for systems with state-dependent switching

In this paper, we consider the existence of quadratic Lyapunov functions for certain types of switched linear systems. Given a partition of the state-space, a set of matrices (linear dynamics), and a matrix-valued function A(x) constructed by associating these matrices with regions of the state-space in a manner governed by the partition, we ask whether there exists a positive definite symmetric matrix P such that A(x)^TP+PA(x) is negative definite for all x(t). For planar systems, necessary and sufficient conditions are given. Extensions for higher order systems are also presented.     

脉冲微分方程正解的存在性

利用锥上的Krasnoselskii不动点定理,证明了二阶非线性具特征值问题的脉冲微分方程正解的存在性.     

脉冲系统输入—输出到状态稳定的Lyapunov条件

在一定的条件下,通过对时间域分割成不同的区间段然后应用Lyapunov函数分析方法给出了脉冲系统输入—输出到状态稳定(IOSS)的一个充分条件,另外,讨论了积分输入—输出到状态稳定的情况.     

Quadratic stabilization of a class of switched nonlinear systems via single Lyapunov function

This paper is concerned with the problem of globally quadratic stabilization for a class of switched cascade systems. The system under consideration is composed of two subsystems: a linear switched part and a nonlinear part, which are also switched systems. The feedback control law and the switching law are designed respectively when the first part is stabilized under some switching law and when both parts can be stabilized under some switching laws. We construct the single Lyapunov functions and design the switching laws based on the structure characteristics of the switched system. Also, the designed switching laws are of hysteresis switching form in order to avoid sliding models.     

Solving Quadratic Equations via PhaseLift When There Are About as Many Equations as Unknowns

This note shows that we can recover any complex vector \(\boldsymbol {x}_{0} \in \mathbb {C}^{n}\) exactly from on the order of n quadratic equations of the form |〈a _i ,x ₀〉|²=b _i , i=1,…,m, by using a semidefinite program known as PhaseLift. This improves upon earlier bounds in Candès et al. (Commun. Pure Appl. Math. 66:1241–1274, 2013), which required the number of equations to be at least on the order of nlogn. Further, we show that exact recovery holds for all input vectors simultaneously, and also demonstrate optimal recovery results from noisy quadratic measurements; these results are much sharper than previously known results.     

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.