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January 2016, 36(1): 97-136. doi: 10.3934/dcds.2016.36.97
Invariance entropy of hyperbolic control sets
Adriano Da Silva 1, and Christoph Kawan 2,
Imecc - Unicamp, Departamento de Matemática, Rua Sérgio Buarque de Holanda, 651, Cidade Universitária Zeferino Vaz 13083-859, Campinas - SP, Brazil
Universität Passau, Fakultät für Informatik und Mathematik, Innstraße 33, 94032 Passau, Germany
Received August 2014 Revised March 2015 Published June 2015
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.
Keywords: control sets, chain control sets, universally regular controls, Invariance entropy, hyperbolicity, shadowing lemma., volume lemma, control-affine systems.
Mathematics Subject Classification: Primary: 93C15, 37D20; Secondary: 93B05, 37C6.
Citation: Adriano Da Silva, Christoph Kawan. Invariance entropy of hyperbolic control sets. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 97-136. doi: 10.3934/dcds.2016.36.97
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Adriano Da Silva Christoph Kawan