Input-to-state stability for impulsive stochastic nonlinear systems with delayed impulses

In this paper, the problems of the stochastic input-to-state stability (SISS) properties are investigated for impulsive stochastic nonlinear systems with delayed impulses. By employing Lyapunov method together with average impulsive interval approach, the sufficient conditions are established to ensure properties for impulsive stochastic systems with all stable subsystems. Moreover, the ISS properties are also derived for the stochastic systems when the system is with both stable and unstable subsystems. Then, we also study impulsive systems with multiple jumps, i.e. there exist different impulses in a system. By now, there has been no papers to consider multiple jumps for stochastic cases. Finally, two examples are provided to illustrate the effectiveness of our results.