Razumikhin-type theorems for ISS of nonlinear delayed impulsive systems

In this paper we investigate the problem of input-to-state stability (ISS) of nonlinear delayed impulsive systems. Both the continuous dynamics and the discrete dynamics of nonlinear impulsive systems are subjected to external input. By virtue of the Razumikhin technique in combination with Lyapunov functions, we obtain some Razumikhin-type theorems that warrant input-to-state stability of nonlinear impulsive systems with time-delays. The Razumikhin-type input-to-state stability theorems cover the cases where the delayed continuous dynamics are input-to-state stable or destabilizing and the discrete dynamics are input-to-state stable or destabilizing, such that nonlinear delayed impulsive systems are able to retain input-to-state stability under certain conditions. The applicability of the derived Razumikhin-type theorems is illustrated by numerical results.