On the existence of nonsmooth control-Lyapunov functions in the sense of generalized gradients

Institut Girard Desargues, Université Claude Bernard Lyon I, 69622 Villeurbanne, France; rifford@desargues.univ-lyon1.fr.

Received: 21 August 2000
Revised: 2 May 2001

Abstract

Let be a general control system; the existence of a smooth control-Lyapunov function does not imply the existence of a continuous stabilizing feedback. However, we show that it allows us to design a stabilizing feedback in the Krasovskii (or Filippov) sense. Moreover, we recall a definition of a control-Lyapunov function in the case of a nonsmooth function; it is based on Clarke's generalized gradient. Finally, with an inedite proof we prove that the existence of this type of control-Lyapunov function is equivalent to the existence of a classical control-Lyapunov function. This property leads to a generalization of a result on the systems with integrator.