ISS characterization of retarded switching systems with relaxed Lyapunov-Krasovskii functionals *

TL;DR

This paper advances the understanding of input-to-state stability in switching retarded systems by establishing new Lyapunov-Krasovskii functional characterizations that are both necessary and sufficient.

Contribution

It introduces relaxed Lyapunov-Krasovskii functionals for ISS characterization, broadening the theoretical framework for analyzing switching retarded systems.

Findings

New necessary and sufficient conditions for ISS using relaxed Lyapunov-Krasovskii functionals.

Characterizations applicable to systems with measurable inputs and switching signals.

Multiple derivative notions explored for ISS conditions.

Abstract

This paper gives further insights about the Lyapunov-Krasovskii characterization of input-tostate stability (ISS) for switching retarded systems on the basis of the results in [I. Haidar and P. Pepe. Lyapunov-krasovskii characterization of the input-to-state stability for switching retarded systems. SIAM Journal on Control and Optimization, 59(4):2997-3016, 2021]. We give new characterizations of the ISS property through the existence of a relaxed common Lyapunov-Krasovskii functional. More precisely, we show that the existence of a continuous Lyapunov-Krasovskii functional whose upper right-hand Dini derivative satisfies a dissipation inequality almost everywhere is necessary and sufficient for the ISS of switching retarded systems with measurable inputs and measurable switching signals. Different characterization results, using different derivative notions, are also given.