Uniform Global Asymptotic Stability for Time-Invariant Delay Systems

Iasson Karafyllis (1) , Pierdomenico Pepe (2) , Antoine Chaillet (3) , Yuan Wang (4)

1 Department of Mathematics, School of Applied Mathematical and Physical Sciences, National Technical University of Athens
2 DISIM - Department of Information Engineering, Computer Science and Mathematics = Dipartimento di Ingegneria e Scienze dell'Informazione e Matematica [L'Aquila]
3 L2S - Laboratoire des signaux et systèmes
4 Florida Atlantic University [Boca Raton]

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 Hölder 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.

Domaines: Automatique / Robotique

Iasson Karafyllis, Pierdomenico Pepe, Antoine Chaillet, Yuan Wang. Uniform Global Asymptotic Stability for Time-Invariant Delay Systems. 61st IEEE Conference on Decision and Control (CDC 2022), Dec 2022, Cancun, Mexico. ⟨10.1109/cdc51059.2022.9992709⟩. ⟨hal-03773601⟩