Stability of interconnected impulsive systems with and without time-delays using Lyapunov methods

TL;DR

This paper investigates the stability of interconnected impulsive systems with and without time-delays using Lyapunov methods, providing conditions for uniform input-to-state stability and tools for constructing Lyapunov functions.

Contribution

It introduces new Lyapunov-based criteria for ISS of impulsive systems with delays and large-scale networks, including construction methods for Lyapunov functions and gains.

Findings

ISS is guaranteed under dwell-time and small-gain conditions.

Lyapunov functions for the entire network can be constructed from subsystem functions.

Theoretical results are demonstrated through illustrative examples.

Abstract

In this paper we consider input-to-state stability (ISS) of impulsive control systems with and without time-delays. We prove that if the time-delay system possesses an exponential Lyapunov-Razumikhin function or an exponential Lyapunov-Krasovskii functional, then the system is uniformly ISS provided that the average dwell-time condition is satisfied. Then, we consider large-scale networks of impulsive systems with and without time-delays and we prove that the whole network is uniformly ISS under a small-gain and a dwell-time condition. Moreover, these theorems provide us with tools to construct a Lyapunov function (for time-delay systems - a Lyapunov-Krasovskii functional or a Lyapunov-Razumikhin function) and the corresponding gains of the whole system, using the Lyapunov functions of the subsystems and the internal gains, which are linear and satisfy the small-gain condition. We illustrate the application of the main results on examples.