On the derivation of stability properties for time-delay systems without constraint on the time-derivative of the initial condition

TL;DR

This paper extends stability analysis methods for time-delay systems to initial conditions that are merely continuous, removing the previous requirement for absolute continuity and square integrability of the initial state derivative.

Contribution

It introduces a novel approach that allows stability results to be applied to less restrictive initial conditions in certain classes of time-delay systems.

Findings

Stability results now hold for continuous initial conditions.

Lyapunov-Krasovskii functionals can be used without derivative constraints.

The approach broadens the applicability of stability analysis methods.

Abstract

Stability of retarded differential equations is closely related to the existence of Lyapunov-Krasovskii functionals. Even if a number of converse results have been reported regarding the existence of such functionals, there is a lack of constructive methods for their selection. For certain classes of time-delay systems for which such constructive methods are lacking, it was shown that Lyapunov-Krasovskii functionals that are also allowed to depend on the time-derivative of the state-trajectory are efficient tools for the study of the stability properties. However, in such an approach the initial condition needs to be assumed absolutely continuous with a square integrable weak derivative. In addition, the stability results hold for initial conditions that are evaluated based on the magnitude of both the initial condition and its time-derivative. The main objective of this paper is to show that, for certain classes of time-delay systems, the aforementioned stability results can actually be extended to initial conditions that are only assumed continuous and that are evaluated in uniform norm.