A metric proof  of the converse Lyapunov theorem for semicontinuous multivalued dynamics

Antonio Siconolfi; Gabriele Terrone

doi:10.3934/dcds.2012.32.4409

Article Contents

2012, Volume 32, Issue 12: 4409-4427. Doi: 10.3934/dcds.2012.32.4409

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A metric proof of the converse Lyapunov theorem for semicontinuous multivalued dynamics

Antonio Siconolfi^1, and
Gabriele Terrone^2,

1.
Dipartimento di Matematica, Università degli Studi di Roma La Sapienza, P.le Aldo Moro 2, 00185 Roma
2.
Instituto Superior Técnico, Universidade Técnica de Lisboa, Departamento de Matemática, Av. Rovisco Pais, 1049-001 Lisboa

Received: January 2011

Revised: June 2012

Published online: August 2012

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Abstract

We extend the metric proof of the converse Lyapunov Theorem, given in [13] for continuous multivalued dynamics, by means of tools issued from weak KAM theory, to the case where the set-valued vector field is just upper semicontinuous. This generality is justified especially in view of application to discontinuous ordinary differential equations. The more relevant new point is that we introduce, to compensate the lack of continuity, a family of perturbed dynamics, obtained through internal approximation of the original one, and perform some stability analysis of it.

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Mathematics Subject Classification: Primary: 37B25, 49L25, 93D05.

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References

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