Input-to-state and integral input-to-state stability for a class of nonlinear time-delay system with delay-dependent impulse effects

This paper deals with the input-to-state stability (ISS) and integral input-to-state stability (iISS) problems of general nonlinear time-delay systems subject to delay-dependent impulse effects. The purpose of these addressed problems is to derive Lyapunov-type criteria for ISS and iISS properties by using the Lyapunov–Krasovskii function method and Halanay-type inequalities. It is shown that when the continuous time-delay systems are ISS (iISS) and the delay-dependent impulse effects are destabilising, the hybrid system is ISS (iISS) provided that the impulsive effects do not occur too frequently. Moreover, the proposed criteria are more general and easier to verify under weaker conditions compared to the existing results. As applications, some examples are utilised to demonstrate the usefulness of the theoretical results.