A superposition approach for the ISS Lyapunov-Krasovskii theorem with pointwise dissipation

TL;DR

This paper introduces a superposition approach to the ISS Lyapunov-Krasovskii theorem, leveraging pointwise dissipation in a Lyapunov functional to establish input-to-state stability with potentially tighter estimates.

Contribution

It develops a new stability theory using a Lyapunov-Krasovskii functional with pointwise dissipation, enhancing stability analysis of time-delay systems.

Findings

The approach ensures input-to-state stability with pointwise dissipation.

The method provides tighter stability estimates.

An example demonstrates the advantages of the approach.

Abstract

We show that the existence of a Lyapunov-Krasovskii functional (LKF) with pointwise dissipation (i.e. dissipation in terms of the current solution norm) suffices for input-to-state stability, provided that uniform global stability can also be ensured using the same LKF. To this end, we develop a stability theory, in which the behavior of solutions is not assessed through the classical norm but rather through a specific LKF, which may provide significantly tighter estimates. We discuss the advantages of our approach by means of an example.