Lyapunov-krasovskii characterization of the input-To-state stability for switching retarded systems

In this paper we characterize various stability notions of nonlinear switching retarded systems by the existence of a common Lyapunov-Krasovskii functional with suitable conditions. We consider a general class of Lebesgue measurable switching signals. We provide an equivalence property showing that uniform input-To-state stability can be equivalently studied through the class of piecewise-constant inputs and piecewise-constant switching signals. Thanks to this equivalence property, we rely on what it is developed in the literature to provide direct and converse theorems for uniform input-To-state, asymptotic, and exponential stability. Based on these results, we give a first-order approximation theorem for nonlinear switching retarded systems. A link between the exponential stability of an unforced switching retarded system and the input-To-state stability property, in the case of measurable switching signals, is obtained. Examples showing the applicability of our results are also given.