Growth conditions for global exponential stability and exp-ISS of time-delay systems under point-wise dissipation

TL;DR

This paper extends the conditions under which global exponential stability and input-to-state stability can be guaranteed for time-delay systems using point-wise dissipation, including robustness considerations.

Contribution

It introduces new conditions for GES and ISS of time-delay systems based on point-wise dissipation, accommodating small history-dependent terms.

Findings

GES and ISS can be ensured with small history-dependent terms

Explicit bounds on the magnitude of additional history terms

Robustness to modeling uncertainties demonstrated through example

Abstract

For time-delay systems, it is known that global asymptotic stability is guaranteed by the existence of a Lyapunov-Krasovskii functional that dissipates in a point-wise manner along solutions, namely whose dissipation rate involves only the current value of the solution's norm. So far, the extension of this result to global exponential stability (GES) holds only for systems ruled by a globally Lipschitz vector field and remains largely open for the input-to-state stability (ISS) property. In this paper, we rely on the notion of exponential ISS to extend the class of systems for which GES or ISS can be concluded from a point-wise dissipation. Our results in turn show that these properties still hold in the presence of a sufficiently small additional term involving the whole state history norm. We provide explicit estimates of the tolerable magnitude of this extra term and show through an example how it can be used to assess robustness with respect to modeling uncertainties.