On Stability in Impulsive Dynamical Systems

Abstract

Summary. Several results on stability in impulsive dynamical systems are proved. The rst main result gives equivalent conditions for stability of a compact set. In particular, a generalization of Ura’s theorem to the case of impulsive systems is shown. The second main theorem says that under some additional assumptions every component of a stable set is stable. Also, several examples indicating possible complicated phenomena in impulsive systems are presented. 1. Preliminaries. Let X be a metric space. A pair (X; ) is a dynamical system if : R X ! X is a continuous function with (0;x) = and (t; (s;x)) = (t +s;x) for every t;s;x; replacingR by R+ we get the denition of a semidynamical system. For the elementary properties of dynamical and semidynamical systems, see [BH], [BS], [NS], [P], [V]. We dene the positive trajectory of as + (x) = ([0; +1);x) = ([0; +1) fxg). In a semidynamical system, for t 0 and y 2 X by F (t;y) we mean fz2 X : (t;z) = yg. In an analogous way we dene F (; D) for [0;1) and D X. A point x2 X is said to be a start point if F (t;x) =; for all t > 0. An impulsive system (X; ;M;I) consists of a semidynamical system (X; ) (which may be a dynamical system), called the basic system, together with a nonempty closed subset M of X and a continuous function I : M ! X. We assume that for each x2 M there is an x > 0 such that (( x; 0);x)\M =; and ((0;x);x)\M =; (for dynamical systems), or F ((0;x);x)\M =; and ((0;x);x)\M =; (for semidynamical systems). These conditions mean that the points of M are isolated on every trajectory 2000 Mathematics Subject Classic ation: Primary 37B25; Secondary 54H20, 34A37.