The ISS framework for time-delay systems: a survey

TL;DR

This survey reviews the tools and methods for analyzing input-to-state stability in time-delay systems, highlighting differences from finite-dimensional systems and discussing stability criteria, interconnected systems, and open questions.

Contribution

It provides a comprehensive overview of ISS concepts, stability analysis techniques, and their applications to interconnected time-delay systems, including open research questions.

Findings

Lyapunov-Krasovskii, Lyapunov-Razumikhin, and Halanay methods for stability

Characterizations of ISS, integral ISS, and input-output stability

Applications to interconnected systems in cascade and feedback

Abstract

At the occasion of Eduardo D. Sontag's 70 th birthday, we provide here an overview of the tools available to study input-to-state stability (ISS) and related notions for time-delay systems. After a hopefully pedagogical presentation of the main differences with respect to the finite-dimensional theory, we review basic stability concepts for input-free time-delay systems, as well as instruments to guarantee them in practice, including the Lyapunov-Krasosvkii, Lyapunov-Razumikhin, and Halanay approaches. We then consider the influence of inputs through the notions of ISS, integral ISS, and input-to-output stability and provide both Lyapunov-like and solutions-based characterizations of these properties. We also show how these notions can be helpful for the stability analysis of interconnected systems, whether in cascade or in feedback form. We finally provide a list of questions which remain open until now.