Lyapunov–Krasovskii‐type criteria on ISS and iISS for impulsive time‐varying delayed systems

Abstract

This study investigates the problem of input-to-state stability (ISS) and integral input-to-state stability (iISS) of impulsive time-varying delayed systems. Some Lyapunov–Krasovskii-type criteria on ISS and iISS which are effective for destabilising impulses and stabilising impulses are derived from advanced linear dissipation inequalities under average impulsive interval constraints. The conditions which require the coefficients of the linear dissipation inequalities on the Lyapunov–Krasovskii functionals to be constants in the existing results on ISS/iISS of impulsive delayed systems are weakened. In this study, with the aid of the notions of uniformly exponentially stable function and average impulsive interval, the results allow the coefficients of the developed linear dissipation inequalities on the Lyapunov–Krasovskii functionals to be time-varying functions which can take both positive and negative values, and the impulsive intervals of an impulsive sequence are allowed to have arbitrarily small lower bound and enough big upper bound simultaneously. Two examples are presented to illustrate the effectiveness of the results.