Remarks on Different Notions on Output Stability for Nonlinear Delay Systems

TL;DR

This paper explores various output stability notions for nonlinear delay systems, clarifying their relationships and providing Razumikhin criteria and Lyapunov theorems to analyze stability in such systems.

Contribution

It extends input-to-state stability concepts to delay systems, establishing relationships among different notions and offering criteria and theorems for output stability analysis.

Findings

Different output stability notions are intrinsically distinct for delay systems.

Razumikhin criteria for various output stability properties are derived.

Converse Lyapunov theorems for delay systems' output stability are developed.

Abstract

Motivated by the regulator theory and adaptive controls, several notions on output stability in the framework of input-to-state stability (iss) were introduced for finite-dimensional systems. It turned out that these output stability notions are intrinsically different, reflecting different manners of how state variables may affect the transient behavior of output variables. In this work, we consider these output stability properties for delay systems. Our main objective is to illustrate how the various notions are related for delay systems and to provide the Razumikhin criteria for the output stability properties. The main results are also critical in developing the converse Lyapunov theorems of the output stability properties for delay systems