Lyapunov's direct method in the research of stability of impulsive systems

In this talk, a system of ordinary differential equations with impulse effect at fixed instants is considered. The system is assumed to have the zero solution. It is shown that the existence of a corresponding Lyapunov function is a necessary and sufficient condition for the uniform asymptotic stability of the zero solution. Restrictions on perturbations of the right-hand sides of differential equations and impulse effect are obtained under which the uniform asymptotic stability of the zero solution of the "unperturbed" system implies the uniform asymptotic stability of the zero solution of the "perturbed" system. In the case of a periodic system with impulse effect, it is shown that if the trivial solution of the system is stable or asymptotically stable, then it is uniformly stable or uniformly asymptotically stable, respectively. By using the method of Lyapunov functions, the criteria of asymptotical stability and instability are obtained.