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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 wherea positive semidefinite function of the solutions satisfies aclass- ${\mathcal{KL}}$ estimatein terms of time and a second positive semidefinite function of theinitial condition.We show that a smooth converse Lyapunov function, i.e., one whosederivative along solutions can beused to establish the class- ${\mathcal{KL}}$ estimate, exists if andonly if the class- ${\mathcal{KL}}$ estimateis robust, i.e., it holds for a larger, perturbed differentialinclusion.It remains an open question whether all class- ${\mathcal{KL}}$ estimates are robust.One sufficient condition for robustness is that the originaldifferential inclusion is locally Lipschitz.Another sufficient condition is that the two positive semidefinitefunctions agree anda backward completability condition holds. These special cases unifyand generalize many resultson converse Lyapunov theorems for differential equations anddifferential inclusions that have appeared in the literature.

Keywords

Differential inclusionsLyapunov functionsuniform asymptotic stability.

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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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