Input-to-State Stability of Time-Delay Systems With Hybrid Impulses and Continuous Subdynamics Based on Vector Lyapunov Function

Abstract

This article focuses on input-to-state stability (ISS) of impulsive time-delay systems where the hybrid effect of impulses with time-dependent multiple jump maps in different subsystems is fully considered. By virtue of M-matrix and vector Lyapunov function, some theorems for ISS are established for Krasovskii-type conditions which avoid the common threshold of impulses in subsystems and allow the simultaneous existence of stable and unstable continuous subdynamics. If the time intervals between impulses are bounded, the ISS of time-delay systems can be ensured even though hybrid impulses exist in each subsystem, which determines the robustness of stable time-delay systems with respect to hybrid impulses. The unstable time-delay systems with different subdynamics can be stabilized in input-to-state-stability sense by involving hybrid impulses. In addition, several extended criteria are presented to bridge the results derived via vector Lyapunov function, scalar Lyapunov function, and comparison principle. At last, the theoretical results are validated by numerical example and a simplified model of sewage treatment tank.