Input-to-State Stability of Impulsive Delay Systems With Multiple Impulses

This article investigates input-to-state stability (ISS) of impulsive delay systems with multiple impulses. We provide sufficient conditions in terms of exponential ISS-Lyapunov functions coupled with Razumikhin-type conditions for establishing the ISS property, where the possibility of the effects of multiple impulses on ISS is fully considered. It shows that when the continuous dynamics are ISS and the discrete dynamics that govern the jumps involve multiple impulses, the ISS property can be retained for arbitrary impulse time sequences if the cumulative strength of multiple impulses satisfies some conditions. Conversely, when the continuous dynamics are not ISS, then the multiple impulses that satisfy some conditions can stabilize the system in ISS sense if there are no overly long intervals between impulses. Two illustrative examples are presented, with their numerical simulations, to demonstrate the main results.