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Journal of Dynamical and Control Systems

, Volume 21, Issue 3, pp 351–377 | Cite as

Fréchet Generalized Trajectories and Minimizers for Variational Problems of Low Coercivity

  • Manuel Guerra
  • Andrey Sarychev
Article

Abstract

We address consecutively two problems. First, we introduce a class of so-called Fréchet generalized controls for a multi-input control-affine system with non-commuting controlled vector fields. For each control of the class, one is able to define a unique generalized trajectory, and the input-to-trajectory map turns out to be continuous with respect to the Fréchet metric. On the other side, the class of generalized controls is broad enough to settle the second problem, which is to prove existence of generalized minimizers of Lagrange variational problem with functionals of low (in particular linear) growth. Besides, we study the possibility of Lavrentiev-type gap between the infima of the functionals in the spaces of ordinary and generalized controls. This is an abridged (due to the journal space limitations) version of a more detailed preprint with several proofs, drawings, and examples added, published in arXiv.org > math > arXiv: 1402.0477.

Keywords

Generalized controls Impulsive controls Non-involutive systems Existence of minimizers Lavrentiev phenomenon 

Mathematics Subject Classifications (2010)

49J15 49N25 93C15 

Notes

Acknowledgments

The research of the first coauthor has been supported by FCT–Fundação para a Ciência e Tecnologia (Portugal) via strategic project PEst-OE/EGE/UI0491/2013; he is grateful to INDAM (Italy) for supporting his visit to the University of Florence in January 2014. The research of the second coauthor has been supported by MIUR (Italy) via national project (PRIN) 200894484E of MIUR (Italy); he is also grateful to CEMAPRE (Portugal) for supporting his research stay at ISEG, University of Lisbon in May 2013.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.CEMAPREUniversity of LisbonLisbonPortugal
  2. 2.ISEGUniversity of LisbonLisbonPortugal

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