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Issue ESAIM: COCV
Volume 5, 2000
Page(s) 313 - 367
DOI 10.1051/cocv:2000113

DOI: 10.1051/cocv:2000113

ESAIM: COCV, June 2000, Vol. 5, 313-367

A smooth Lyapunov function from a class- ${\mathcal}$ estimate involving two positive semidefinite functions[*]

Andrew R. Teel
ECE Department, University of California, Santa Barbara, CA 93106, U.S.A.; (teel@ece.ucsb.edu)

Laurent Praly
Centre Automatique et Systèmes, École des Mines de Paris, 35 rue Saint Honoré, 77305 Fontainebleau Cedex, France; (praly@cas.ensmp.fr)

Received November 3, 1999. Revised May 24, 2000

Abstract: We consider differential inclusions where a positive semidefinite function of the solutions satisfies a class- ${\mathcal}$ 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- ${\mathcal}$ estimate, exists if and only if the class- ${\mathcal}$ estimate is robust, i.e., it holds for a larger, perturbed differential inclusion. It remains an open question whether all class- ${\mathcal}$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.

Keywords and phrases: Differential inclusions, Lyapunov functions, uniform asymptotic stability.

AMS Subject Classification: 34A60, 34D20, 34B25.

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