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; email@example.com.
Revised: 2 May 2001
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.
Mathematics Subject Classification: 93D05 / 93D15 / 93D20 / 93D30 / 93D09 / 93B05
Key words: Asymptotic stabilizability / converse Lyapunov theorem / nonsmooth analysis / differential inclusion / and Krasovskii solutions / feedback.
© EDP Sciences, SMAI, 2001