Lyapunov Characterization of Input-to-State Stability for a Class of Impulsive Systems

We prove that the existence of an input-to-state stability (ISS)-Lyapunov function is a necessary and sufficient condition for the input-to-state stability of a class of impulsive systems over Banach spaces. A key feature of our work is that it employs time-varying ISS-Lyapunov functions as opposed to candidate ISS-Lyapunov functions, which also makes it possible to conclude ISS of impulsive systems with unstable continuous and discrete dynamics.