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A smooth Lyapunov function from a class-${\mathcal{KL}}$ estimateinvolving two positive semidefinite functions

Published online by Cambridge University Press:  15 August 2002

Andrew R. Teel  and
Laurent Praly
Andrew R. Teel
Affiliation:
ECE Department, University of California, Santa Barbara, CA 93106, U.S.A.; teel@ece.ucsb.edu.
Laurent Praly
Affiliation:
Centre Automatique et Systèmes, École des Mines de Paris, 35 rue Saint Honoré, 77305 Fontainebleau Cedex, France; praly@cas.ensmp.fr.
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Abstract

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

Differential inclusionsLyapunov functionsuniform asymptotic stability.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 5 , 2000 , pp. 313 - 367
Copyright
© EDP Sciences, SMAI, 2000

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