Lyapunov small-gain theorems for networks of not necessarily ISS hybrid systems

TL;DR

This paper introduces a new Lyapunov-based small-gain theorem applicable to networks of hybrid systems that may not be input-to-state stable, extending existing results and incorporating average dwell-time clocks to broaden applicability.

Contribution

It presents a novel small-gain theorem for hybrid system networks that do not require ISS, unifying and extending previous hybrid and impulsive system results.

Findings

Unified small-gain theorem for hybrid systems

Incorporation of average dwell-time clocks to modify Lyapunov functions

Broader applicability to non-ISS hybrid system networks

Abstract

We prove a novel Lyapunov-based small-gain theorem for networks of $ n \geq 2 $ hybrid systems which are not necessarily input-to-state stable. This result unifies and extends several small-gain theorems for hybrid and impulsive systems proposed in the last few years. We also show how average dwell-time (ADT) clocks and reverse ADT clocks can be used to modify the Lyapunov functions for subsystems and to enlarge the applicability of the derived small-gain theorems.