Is Global Asymptotic Stability Necessarily Uniform for Time-Delay Systems?

TL;DR

This paper explores the relationship between global asymptotic stability and uniform global asymptotic stability in time-invariant delay systems, establishing conditions under which they are equivalent and providing new Lyapunov characterizations.

Contribution

It demonstrates the equivalence of GAS and UGAS for delay systems under robust forward completeness and specific function space conditions, with novel Lyapunov characterizations.

Findings

GAS and UGAS are equivalent under robust forward completeness.

Equivalence holds in Sobolev and Holder spaces.

Provides new Lyapunov criteria for stability.

Abstract

For time-invariant finite-dimensional systems, it is known that global asymptotic stability (GAS) is equivalent to uniform global asymptotic stability (UGAS), in which the decay rate and transient overshoot of solutions are requested to be uniform on bounded sets of initial states. This paper investigates this relationship for time-invariant delay systems. We show that UGAS and GAS are equivalent for this class of systems under the assumption of robust forward completeness, i.e. under the assumption that the reachable set from any bounded set of initial states on any finite time horizon is bounded. We also show that, if the state space is a space in a particular family of Sobolev or Holder spaces, then GAS is equivalent to UGAS and that robust forward completeness holds. Based on these equivalences, we provide a novel Lyapunov characterization of GAS (and UGAS) in the aforementioned spaces.