Generalized Lyapunov method for discontinuous systems

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Abstract

Nonlinear dynamical systems described by differential equations with discontinuous right-hand side and solutions intended in Filippov sense are considered. Based on Filippov differential inclusion and a new chain rule for differentiating regular functions along Filippov solution trajectories, different kinds of stability and convergence results are presented. Moreover, we investigate the stability and convergence for the corresponding perturbation system of the discontinuous system and some new criteria are addressed.

Introduction

Nonlinear dynamical systems described by differential equations with discontinuous right-hand side (discontinuous systems) have been widely studied in recent years, and are frequently found in controllability problems, stabilization as well as optimal control problems (see, e.g. [1], [2], [3], [4], [5]). Due to the discontinuity of the right-hand side of systems, many results in the classical theory of differential systems have been shown to be invalid for discontinuous systems. Therefore, the first theoretical problem of giving an appropriate definition of solution for the discontinuous system, emerged. Different methods have been presented in the literature [6], [7], [8]. In 1964, Filippov [6] developed a solution concept for differential systems whose right-hand side only required to be Lebesgue measurable in the state and time variables. The notion of Filippov solution has been accepted universally as a good one for discontinuous systems. Based on the Filippov solution, the results were presented systematically for existence, uniqueness, and continuous dependence on initial conditions in [6]. However, the issue of the stability analysis is still incomplete.

Stability analysis of nonlinear system is often based on Lyapunov methods, which was firstly proposed by Lyapunov and has been enduringly generalized. In [9], Yoshizawa developed the Lyapunov theory for Lipschitz continuous potential functions, but the right-hand side of the system is assumed continuous and the solution trajectories are assumed smooth. For discontinuous Carathéodory systems, recent papers [10], [11], [12] presented some modifications of Lyapunov method, but the right-hand side of the systems are continuous in state variable. In [6], Filippov investigated the stability of discontinuous system, but dealt with smooth Lyapunov functions. For Filippov solution, Lyapunov method have been widely developed (see [13], [14], [15], [16], [17], [18]). However, in [13], [14], [15], [16], [18], the systems considered are autonomous and the Lyapunov functions are not explicitly dependent of time variable. In [13], [14], [15], [16], [17], global exponential stability, global exponential convergence, and convergence in finite time were not investigated. In this paper, we consider non-smooth Lyapunov functions, which are local Lipschitz continuous and regular functions. Resorting to non-smooth Lyapunov functions gives more flexibility in the stability analysis for discontinuous Filippov systems.

The paper is organized as follows: In Section 2, some fundamental tools needed in order to obtain the main results are collected, such as the definition of Filippov solution, Clark generalized gradient and the chain rule. Section 3 presents different kinds of stability and convergence criteria for discontinuous Filippov systems. In Section 4, the perturbation problem for discontinuous Filippov systems is firstly proposed, then under certain limits on the perturbations, the sufficient conditions for conservation of stability and convergence are obtained. Finally, some conclusions are stated in Section 5.

Section snippets

Preliminaries

First of all, we introduce the concept of Filippov solution [6].

Consider the following non-autonomous differential system ẋ=f(t,x), where f:R×RnRn is measurable and essentially locally bounded.

Let F(t,x) be the smallest closed convex set containing all limit values of the vector function f(t,x), where x tending to x, x spans almost the whole neighborhood (that is, except for a set of measure zero) of the point x, that is, F(t,x)=δ>0μN=0co¯f(t,B(x,δ)N). Here B(x,δ) is the ball of center x

Convergence and stability theorems

In this section, we present some convergence and stability results for system (1) by means of a regular Lyapunov functions.

In this paper, the Lyapunov functions we considered have the following properties.

(H0)V:Rn×RR is a regular function satisfying V(t,0)=0 for any tR, and there exists a continuous non-decreasing function u:R+1R+1 such that V(x,t)u(x), where u(s) is positive for s>0, u(0)=0 and u(s)+ as s+.

Note that a Lyapunov function V(x,t) satisfying (H0) need not to be

Robustness with respect to perturbations of f

In this section, we shall prove that if system (1) is stable (or convergent), then it is possible to produce a specific kind of “inflation” of f and the perturbation system is still stable (or convergent). In this sense, the stability (or convergence) is robust with respect to a certain class of perturbations.

The perturbation system of system (1) considered in this paper has the following form: ẋ=f(t,x)+δ(t,x), where f,δ:R×RnRn are measurable and essentially locally bounded.

Remark 5

The perturbation δ(

Conclusion

In this paper, nonlinear dynamical systems described by differential equations with discontinuous right-hand side and solutions intended in Filippov sense have been considered. The interest in such equations is motivated by the fact that they naturally arise in stabilization problems in connection with important classes of discontinuous feedback laws. By exploiting the Filippov solution concept and the Clarke generalized gradient, we have generalized the stability and convergence theorems to

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  • Cited by (0)

    Research supported by National Natural Science Foundation of China (10771055, 60835004) and Key Program of Application Science Foundation of Hunan Province (2008FJ2008).

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