On Liapunov-Krasovskii Functionals under Carathéodory Conditions

In [Driver, R. D. (1962). Existence and stability of solutions of a delay-differential system. Archive for Rational Mechanics and Analysis 10, 401–426] a proper definition, not involving the solution, of the derivative of the Liapunov–Krasovskii functional for retarded functional differential equations with continuous right side is given and it is showed that this definition coincides with the non-constructive one given in Krasovskii [1956. On the application of the second method of A. M. Lyapunov to equations with time delays (in Russian). Prikladnaya Matematika i Mekhanika 20, 315–327] involving the solution, for functionals V which are locally Lipschitz (and not only continuous, as it is considered in most literature). In this paper, the result by Driver is extended to a general class of retarded functional differential equations coupled with continuous time difference equations with more general right sides, verifying the Carathéodory conditions. Such result is applied to build a new Liapunov–Krasovskii theorem for studying the input-to-state stability of time-invariant neutral functional differential equations with linear difference operator. An example taken from the literature, concerning transmission lines, is reported, showing the effectiveness of the methodology.