Smooth Lyapunov Functions for Discontinuous Stable Systems

It has been proved that a differential system d x / d t = f(t, x) with a discontinuous right-hand side admits some continuous weak Lyapunov function if and only if it is robustly stable. This paper focuses on the smoothness of such a Lyapunov function. An example of an (asymptotically) stable system for which there does not exist any (even weak) Lyapunov functions of class C ¹ is given. In the more general context of differential inclusions, the existence of a weak Lyapunov function of class C ¹ (or C ) is shown to be equivalent to the robust stability of some perturbed system obtained in introducing measurement error with respect to x and t. This condition is proved to be satisfied by most of the robustly stable systems encountered in the literature. Analogous results are given for the Lagrange stability. As an application to the study of the links between internal and external stability for control systems, an extension of a result by Bacciotti and Beccari is obtained by means of a smooth Lyapunov function associated with a robustly Lagrange stable system.