On the Derivation of Stability Properties for Time-Delay Systems Without Constraint on the Time-Derivative of the Initial Condition

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 article 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.

Automated summary

The stability of retarded differential equations is closely related to the existence of Lyapunov-Krasovskii functionals, but there is a lack of constructive methods for their selection. For certain classes of time-delay systems, Lyapunov-Krasovskii functionals that depend on the time-derivative of the state-trajectory have been shown to be efficient tools for studying stability properties. The main objective of the article is to extend the stability results to initial conditions that are only assumed continuous and evaluated in uniform norm for certain classes of time-delay systems.