# American Institute of Mathematical Sciences

December  2012, 32(12): 4409-4427. doi: 10.3934/dcds.2012.32.4409

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

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

Received  January 2011 Revised  June 2012 Published  August 2012

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.
Citation: Antonio Siconolfi, Gabriele Terrone. A metric proof of the converse Lyapunov theorem for semicontinuous multivalued dynamics. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4409-4427. doi: 10.3934/dcds.2012.32.4409
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