Asymptotic stability and smooth Lyapunov functions for a class of abstract dynamical systems

Michael Schönlein

doi:10.3934/dcds.2017172

Article Contents

2017, Volume 37, Issue 7: 4053-4069. Doi: 10.3934/dcds.2017172

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Asymptotic stability and smooth Lyapunov functions for a class of abstract dynamical systems

Michael Schönlein

Institute for Mathematics, University of Würzburg, Emil-Fischer Straβe 40, 97074 Würzburg, Germany

Received: June 2015

Revised: February 2017

Published: August 2017

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Abstract

This paper deals with a characterization of asymptotic stability for a class of dynamical systems in terms of smooth Lyapunov pairs. We point out that well known converse Lyapunov results for differential inclusions cannot be applied to this class of dynamical systems. Following an abstract approach we put an assumption on the trajectories of the dynamical systems which demands for an estimate of the difference between trajectories. Under this assumption, we prove the existence of a $C^∞$-smooth Lyapunov pair. We also show that this assumption is satisfied by differential inclusions defined by Lipschitz continuous set-valued maps taking nonempty, compact and convex values.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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