Stability criteria of impulsive differential systems☆
Introduction
The mathematical theory of impulsive differential equations has been developed by a large number of mathematicians, see e.g. [1], [2], [3], and their studies have attracted much attention. Furthermore they have been successful in different approaches based on Lyapunov's direct method and comparison technique [2]. In recent years the study of such systems and applications of theory for impulsive differential system have been very intensive (see [4], [5], [6], [7], [8], [9], [10], [11] and their bibliographies). In paper [4], the author has extended the notion of eventual stability to impulsive systems of differential equations, some criteria and results are given. The paper [5] has extended this notion to group of impulsive differential equations. In paper [6], we have obtained some less conservative conditions for asymptotic stability of impulsive differential systems by using new comparison theorems.
In this paper, we shall also consider the uniform eventual asymptotic stability and uniform eventual Lipschitz stability of impulsive differential systems. We obtain new comparison theorems about uniform eventual asymptotic stability and uniform eventual Lipschitz stability of impulsive differential systems, respectively. Compared with the existing result [4], the condition of the eventual stability for impulsive differential systems with impulses at fixed times is less conservative. Our technique depends on Lyapunov's direct method.
The paper is organized as follows. In Section 2, we introduce some preliminary definitions and result which will be used throughout the paper. In Section 3, new comparison theorem and sufficient condition for the uniform eventual asymptotic stability and uniform eventual Lipschitz stability of impulsive differential systems with impulses at fixed times are given.
Section snippets
Preliminaries
An impulsive differential system with impulses at fixed times is described bywhere f:R+×Rn→Rn is continuous; Ik:Rn→Rn is continuous; X∈Rn is the state variable; 0<τ1<τ2<⋯<τk<τk+1<⋯, τk→∞ as k→∞. Definition 1 Let V:R+×Rn→R+, then V is said to belong to the class V0 if V is continuous in (τk−1,τk]×Rn and for each X∈Rn, k=1,2,…, lim(t,Y)→(τk+,X)V(t,Y)=V(τk+,X) exists; V is locally Lipschitzian in X.[1]
Definition 2 [1]
For (t,X)∈(τi−1,τi]×Rn we define
Main results
In this section, we discuss the uniform eventual Lipschitz stability and uniform eventual asymptotic stability of impulsive differential system. Theorem 1 Assume that the following three conditions V:R+×Rn→R+, V∈V0, K(t)D+(t,X)+D+K(t)V(t,X)⩽g(t,K(t)V(t,X)), t≠τk, where g(t,0)=0, and g is continuous in (τk−1, τk]×Rn for each x∈Rn, k=1,2,…, lim(t,y)→(τk+,x)g(t,y)= g(τk+,x) exists, K(t) is a bounded function and K(t)⩾m>0, limt→τk−K(t)=K(τk), i.e. K(t) is left continuous at t=τk, limt→τk+K(t) exists, k=1,2,…,
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Cited by (14)
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2007, Physics Letters, Section A: General, Atomic and Solid State PhysicsCitation Excerpt :For example, the predictive Poincaré control [4] and the occasional proportional feedback control [5] are two impulsive control schemes with varying impulsive intervals. Li et al. in [6] and Sun et al. in [7,8] given some less conservative conditions of asymptotic stability for impulsive systems and the results have been used to design impulsive control for a class of nonlinear systems, respectively. Yang et al. investigated the stabilization and synchronization of a class of chaotic systems called Lorenz systems in [9].
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2007, Journal of Mathematical Analysis and ApplicationsBoundary value problem of second order impulsive functional differential equations
2006, Journal of Mathematical Analysis and ApplicationsNonlinear boundary value problem of first order impulsive functional differential equations
2006, Journal of Mathematical Analysis and Applications
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This work was supported by Shanghai City Natural Science Foundation of China.