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

Hugo Lhachemi (1, 2) , Robert Shorten (3)

1 L2S - Laboratoire des signaux et systèmes
2 CentraleSupélec
3 Imperial College London

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.

Mots clés: Time-delay systems; retarded differential equations; Lyapunov-Krasovskii functionals; stability properties; regularity of the initial condition

Domaines: Optimisation et contrôle [math.OC]; Automatique

Hugo Lhachemi, Robert Shorten. On the derivation of stability properties for time-delay systems without constraint on the time-derivative of the initial condition. IEEE Transactions on Automatic Control, 2020, 66 (11), pp.1558-2523. ⟨10.1109/TAC.2020.3047358⟩. ⟨hal-03084494⟩