Input-to-state stability of stochastic differential systems with hybrid delay-dependent impulses

This paper discusses the pth moment input-to-state stability(p-ISS) and pth moment integral input-to-state stability(p-iISS) for impulsive stochastic systems with time-varying delays and hybrid delayed impulses. With the Lyapunov method and some techniques, the criteria for the impulsive stochastic system p-ISS and p-iISS are established under two common hypotheses, while our criterion allows the time delay to be less than, equal to, or greater than the length of the impulsive interval. The results show that time delay, intensity, and density of hybrid delayed impulses are the main factors affecting the p-ISS and p-iISS of the system; In other words, changing one or more of these factors can stabilize the unstable system. In addition, if both delayed impulses and continuous dynamics are stable, then time delay does not affect the stability of impulsive stochastic systems. The impulsive systems may be p-ISS and p-iISS when unstable delayed impulses perturb the stable continuous dynamics. Finally, two examples are presented to show the validity and correctness of results.