We prove that the existence of an 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.
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Bachmann et al. (2024) studied this question.
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