On the stability of invariant sets of systems with impulse effect

A system of differential equations with impulse effect is considered. It is assumed that this system has an invariant set M. By means of the direct Lyapunov method, the necessary and sufficient conditions of its uniform asymptotic stability are obtained. The conditions on the perturbations of right hand sides of differential equations and impulse effects, under which the uniform asymptotic stability of the invariant set M of the nonperturbed system implies the uniform asymptotic stability of the invariant set of the perturbed system, are obtained. The stability properties of invariant sets of periodic systems are also Studied. (C) 2007 Elsevier Ltd. All rights reserved.