A smooth Lyapunov function from a class- estimate involving two positive semidefinite functions

¹ ECE Department, University of California, Santa Barbara, CA 93106, U.S.A.; teel@ece.ucsb.edu.
² Centre Automatique et Systèmes, École des Mines de Paris, 35 rue Saint Honoré, 77305 Fontainebleau Cedex, France; praly@cas.ensmp.fr.

Received: 3 November 1999
Revised: 24 May 2000

Abstract

We consider differential inclusions where a positive semidefinite function of the solutions satisfies a class- estimate in terms of time and a second positive semidefinite function of the initial condition. We show that a smooth converse Lyapunov function, i.e., one whose derivative along solutions can be used to establish the class- estimate, exists if and only if the class- estimate is robust, i.e., it holds for a larger, perturbed differential inclusion. It remains an open question whether all class- estimates are robust. One sufficient condition for robustness is that the original differential inclusion is locally Lipschitz. Another sufficient condition is that the two positive semidefinite functions agree and a backward completability condition holds. These special cases unify and generalize many results on converse Lyapunov theorems for differential equations and differential inclusions that have appeared in the literature.