New Criteria on $p$th Moment Input-to-State Stability of Impulsive Stochastic Delayed Differential Systems

By employing the Razumikhin technique, the uniformly exponentially stable function and the average dwell-time (ADT) approach, some Razumikhin-type theorems on input-to-state stability (ISS) for time-varying impulsive stochastic delay differential systems (ISDDS) are obtained. It is shown that if the time-varying continuous stochastic delayed dynamics is ISS and the impulsive effects are destabilizing, then the ISDDS is input-to-state stable (ISS) with respect to a lower bound of the average dwell-time (ADT). The existing results on ISS of impulsive systems in the literature require the coefficients of the estimated upper bound for the diffusion operator of a Lyapunov function to be constant numbers. While, the results in this note allow us the coefficients of the estimated upper bound of the diffusion operator to be sign-changing time-varying function. An example is given to illustrate the effectiveness of our results. From the example, one can see the existing results fail to be used to determine the ISS for some type of ISDDS.