On Input-to-State Stability of Impulsive Stochastic Systems with Time Delays

Abstract

This paper is concerned withpth moment input-to-state stability (p-ISS) and stochastic input-to-state stability (SISS) of impulsive stochastic systems with time delays. Razumikhin-type theorems ensuringp-ISS/SISS are established for the mentioned systems with external input affecting both the continuous and the discrete dynamics. It is shown that when the impulse-free delayed stochastic dynamics arep-ISS/SISS but the impulses are destabilizing, thep-ISS/SISS property of the impulsive stochastic systems can be preserved if the length of the impulsive interval is large enough. In particular, if the impulse-free delayed stochastic dynamics arep-ISS/SISS and the discrete dynamics are marginally stable for the zero input, the impulsive stochastic system isp-ISS/SISS regardless of how often or how seldom the impulses occur. To overcome the difficulties caused by the coexistence of time delays, impulses, and stochastic effects, Razumikhin techniques and piecewise continuous Lyapunov functions as well as stochastic analysis techniques are involved together. An example is provided to illustrate the effectiveness and advantages of our results.