On Liapunov-Krasovskii functionals under Carathéodory conditions

In this paper, a basic property that a Liapunov-Krasovskii functional must have when used for studying the stability of time delay systems under Carath¶eodory conditions is studied. This property concerns the equality, almost everywhere, of the upper right-hand Dini derivative of the Liapunov-Krasovskii functional evaluated at the solution, and of the Liapunov-Krasovskii derivative (computed by using everywhere the system dynamics) evaluated at the solution and at the forcing input. Su±cient conditions for that property are provided. Copyright @ 2005 IFAC.