Invariance entropy of hyperbolic control sets

Adriano Da  Silva; Christoph  Kawan

doi:10.3934/dcds.2016.36.97

Article Contents

2016, Volume 36, Issue 1: 97-136. Doi: 10.3934/dcds.2016.36.97

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Invariance entropy of hyperbolic control sets

Adriano Da Silva^1, and
Christoph Kawan^2,

1.
Imecc - Unicamp, Departamento de Matemática, Rua Sérgio Buarque de Holanda, 651, Cidade Universitária Zeferino Vaz 13083-859, Campinas - SP
2.
Universität Passau, Fakultät für Informatik und Mathematik, Innstraße 33, 94032 Passau

Received: July 31, 2014

Revised: February 28, 2015

Published: May 2017

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Abstract

In this paper, we improve the known estimates for the invariance entropy of a nonlinear control system. For sets of complete approximate controllability we derive an upper bound in terms of Lyapunov exponents and for uniformly hyperbolic sets we obtain a similar lower bound. Both estimates can be applied to hyperbolic chain control sets, and we prove that under mild assumptions they can be merged into a formula. The proof of our result reveals the interesting qualitative statement that there exists no control strategy to make a uniformly hyperbolic chain control set invariant that cannot be beaten or at least approached (in the sense of lowering the necessary data rate) by the strategy to stabilize the system at a periodic orbit in the interior of this set.

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Mathematics Subject Classification: Primary: 93C15, 37D20; Secondary: 93B05, 37C60.

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