Method of Lyapunov functions in problems of stability of solutions of systems of differential equations with impulse action

A system of ordinary differential equations with impulse action at fixed moments of time 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 actions 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.