Pth moment input

This paper investigates the pth moment input-to-state stability (ISS) and the pth moment integral input-to-state stability (iISS) of stochastic impulsive switched delayed system with delayed impulses. By employing the method of multiple Lyapunov-Krasovskii functionals and the uniformly exponentially stable function, some relaxed Krasovskii-type sufficient conditions ensuring the pISS/piISS of the addressed systems are developed. These conditions imply the relationship among the impulsive frequency, the time delay existing in impulses, and the coefficients of the estimated upper bound for the derivative of a Lyapunov function. It is shown that if the continuous stochastic delayed dynamics is ISS, and the impulsive effects are destabilizing, then the stochastic impulsive switched delayed system is ISS with respect to the relationship. Compared with the existing results, the conditions obtained results have three relaxations, that is, the derivative of Lyapunov functions of subsystems are allowed to be sign-changing time-varying function rather than a negative definite constant, all subsystems are allowed to be unstable, and the effect of delayed impulses are considered. Finally, an example is provided to illustrate the effectiveness of the results