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.