Razumikhin and Krasovskii methods for asymptotic stability of nonlinear delay impulsive systems on time scales

Abstract

Abstract In this paper, asymptotic stability problems of nonlinear delay impulsive systems on time scales are considered based on the Razumikhin and Krasovskii methods. First, a Krasovskii-type theorem is provided for uniform asymptotic stability and uniform exponential stability of delay impulsive systems on time scales. It is shown that this Krasovskii stability criterion does not require the time derivative of the Lyapunov functional to be necessarily nonpositive on each impulsive interval. Second, asymptotic stability problems of delay impulsive systems on time scales is investigated based on the Razumikhin approach. The advantage of this method is that the length of each impulse interval does not depend on the time delay, which results in the fact that the state trajectory may not decrease instantly and sharply at each impulsive point. Moreover, the idea of this paper provides a unified approach to study continuous system and its discrete counterpart simultaneously. An example is given for illustrating the effectiveness of the proposed results.