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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,

1. 

Imecc - Unicamp, Departamento de Matemática, Rua Sérgio Buarque de Holanda, 651, Cidade Universitária Zeferino Vaz 13083-859, Campinas - SP, Brazil
2. 

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, 2016, 36 (1) : 97-136. doi: 10.3934/dcds.2016.36.97
References:
[1]

E. Akin, Simplicial dynamical systems, Mem. Amer. Math. Soc., 140 (1999), x+197 pp. doi: 10.1090/memo/0667.  Google Scholar

[2]

V. A. Boichenko, G. A. Leonov and V. Reitmann, Dimension Theory for Ordinary Differential Equations, Teubner, Stuttgart, 2005. doi: 10.1007/978-3-322-80055-8.  Google Scholar

[3]

R. Bowen, Topological entropy and axiom $A$, Global Analysis, Proc. Sympos. Pure Math., 14 (1970), 23-41.  Google Scholar

[4]

R. Bowen, Periodic orbits for hyperbolic flows, Amer. J. Math., 94 (1972), 1-30. doi: 10.2307/2373590.  Google Scholar

[5]

F. Colonius and W. Du, Hyperbolic control sets and chain control sets, J. Dynam. Control Systems, 7 (2001), 49-59. doi: 10.1023/A:1026645605711.  Google Scholar

[6]

F. Colonius and C. Kawan, Invariance entropy for control systems, SIAM J. Control Optim., 48 (2009), 1701-1721. doi: 10.1137/080713902.  Google Scholar

[7]

F. Colonius and W. Kliemann, The Dynamics of Control, Birkhäuser, Boston, 2000. doi: 10.1007/978-1-4612-1350-5.  Google Scholar

[8]

F. Colonius and W. Kliemann, Dynamical Systems and Linear Algebra, Graduate Studies in Mathematics, 158, AMS, 2014.  Google Scholar

[9]

J.-M. Coron, Linearized control systems and applications to smooth stabilization, SIAM J. Control Optim., 32 (1994), 358-386. doi: 10.1137/S0363012992226867.  Google Scholar

[10]

A. Da Silva and C. Kawan, Hyperbolic chain control sets on flag manifolds, preprint, submitted (Nov. 2014), arXiv:1402.5841. Google Scholar

[11]

M. F. Demers and L.-S. Young, Escape rates and conditionally invariant measures, Nonlinearity, 19 (2006), 377-397. doi: 10.1088/0951-7715/19/2/008.  Google Scholar

[12]

C. Kawan, Invariance entropy of control sets, SIAM J. Control Optim., 49 (2011), 732-751. doi: 10.1137/100783340.  Google Scholar

[13]

C. Kawan, Invariance Entropy for Deterministic Control Systems - An Introduction, Lecture Notes in Mathematics, 2089, Springer-Verlag, Berlin, 2013. doi: 10.1007/978-3-319-01288-9.  Google Scholar

[14]

C. Kawan and T. Stender, Growth rates for semiflows on Hausdorff spaces, J. Dynam. Differential Equations, 24 (2012), 369-390. doi: 10.1007/s10884-012-9242-9.  Google Scholar

[15]

O. S. Kozlovski, An integral formula for topological entropy of $\CC^{\infty}$ maps, Ergodic Theory Dynam. Systems, 18 (1998), 405-424. doi: 10.1017/S0143385798100391.  Google Scholar

[16]

P.-D. Liu, Random perturbations of axiom $A$ basic sets, J. Stat. Phys., 90 (1998), 467-490. doi: 10.1023/A:1023280407906.  Google Scholar

[17]

G. N. Nair, R. J. Evans, I. M. Y. Mareels and W. Moran, Topological feedback entropy and nonlinear stabilization, IEEE Trans. Automat. Control, 49 (2004), 1585-1597. doi: 10.1109/TAC.2004.834105.  Google Scholar

[18]

H. Nijmeijer and A. van der Schaft, Nonlinear Dynamical Control Systems, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4757-2101-0.  Google Scholar

[19]

M. Qian and Z. Zhang, Ergodic theory for axiom $A$ endomorphisms, Ergod. Th. & Dynam. Sys., 15 (1995), 161-174. doi: 10.1017/S0143385700008294.  Google Scholar

[20]

L. A. B. San Martin and L. Seco, Morse and Lyapunov spectra and dynamics on flag bundles, Ergod. Th. & Dynam. Sys., 30 (2010), 893-922. doi: 10.1017/S0143385709000285.  Google Scholar

[21]

E. D. Sontag, Finite-dimensional open-loop control generators for nonlinear systems, International J. Control, 47 (1988), 537-556. doi: 10.1080/00207178808906030.  Google Scholar

[22]

E. D. Sontag, Mathematical Control Theory. Deterministic Finite-Dimensional Systems, $2^{nd}$ edition, Texts in Applied Mathematics, 6. Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0577-7.  Google Scholar

[23]

Y. Yomdin, Volume growth and entropy, Israel J. Math., 57 (1987), 285-300. doi: 10.1007/BF02766215.  Google Scholar

[24]

L.-S. Young, Large deviations in dynamical systems, Trans. Amer. Math. Soc., 318 (1990), 525-543. doi: 10.2307/2001318.  Google Scholar

show all references

References:
[1]

E. Akin, Simplicial dynamical systems, Mem. Amer. Math. Soc., 140 (1999), x+197 pp. doi: 10.1090/memo/0667.  Google Scholar

[2]

V. A. Boichenko, G. A. Leonov and V. Reitmann, Dimension Theory for Ordinary Differential Equations, Teubner, Stuttgart, 2005. doi: 10.1007/978-3-322-80055-8.  Google Scholar

[3]

R. Bowen, Topological entropy and axiom $A$, Global Analysis, Proc. Sympos. Pure Math., 14 (1970), 23-41.  Google Scholar

[4]

R. Bowen, Periodic orbits for hyperbolic flows, Amer. J. Math., 94 (1972), 1-30. doi: 10.2307/2373590.  Google Scholar

[5]

F. Colonius and W. Du, Hyperbolic control sets and chain control sets, J. Dynam. Control Systems, 7 (2001), 49-59. doi: 10.1023/A:1026645605711.  Google Scholar

[6]

F. Colonius and C. Kawan, Invariance entropy for control systems, SIAM J. Control Optim., 48 (2009), 1701-1721. doi: 10.1137/080713902.  Google Scholar

[7]

F. Colonius and W. Kliemann, The Dynamics of Control, Birkhäuser, Boston, 2000. doi: 10.1007/978-1-4612-1350-5.  Google Scholar

[8]

F. Colonius and W. Kliemann, Dynamical Systems and Linear Algebra, Graduate Studies in Mathematics, 158, AMS, 2014.  Google Scholar

[9]

J.-M. Coron, Linearized control systems and applications to smooth stabilization, SIAM J. Control Optim., 32 (1994), 358-386. doi: 10.1137/S0363012992226867.  Google Scholar

[10]

A. Da Silva and C. Kawan, Hyperbolic chain control sets on flag manifolds, preprint, submitted (Nov. 2014), arXiv:1402.5841. Google Scholar

[11]

M. F. Demers and L.-S. Young, Escape rates and conditionally invariant measures, Nonlinearity, 19 (2006), 377-397. doi: 10.1088/0951-7715/19/2/008.  Google Scholar

[12]

C. Kawan, Invariance entropy of control sets, SIAM J. Control Optim., 49 (2011), 732-751. doi: 10.1137/100783340.  Google Scholar

[13]

C. Kawan, Invariance Entropy for Deterministic Control Systems - An Introduction, Lecture Notes in Mathematics, 2089, Springer-Verlag, Berlin, 2013. doi: 10.1007/978-3-319-01288-9.  Google Scholar

[14]

C. Kawan and T. Stender, Growth rates for semiflows on Hausdorff spaces, J. Dynam. Differential Equations, 24 (2012), 369-390. doi: 10.1007/s10884-012-9242-9.  Google Scholar

[15]

O. S. Kozlovski, An integral formula for topological entropy of $\CC^{\infty}$ maps, Ergodic Theory Dynam. Systems, 18 (1998), 405-424. doi: 10.1017/S0143385798100391.  Google Scholar

[16]

P.-D. Liu, Random perturbations of axiom $A$ basic sets, J. Stat. Phys., 90 (1998), 467-490. doi: 10.1023/A:1023280407906.  Google Scholar

[17]

G. N. Nair, R. J. Evans, I. M. Y. Mareels and W. Moran, Topological feedback entropy and nonlinear stabilization, IEEE Trans. Automat. Control, 49 (2004), 1585-1597. doi: 10.1109/TAC.2004.834105.  Google Scholar

[18]

H. Nijmeijer and A. van der Schaft, Nonlinear Dynamical Control Systems, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4757-2101-0.  Google Scholar

[19]

M. Qian and Z. Zhang, Ergodic theory for axiom $A$ endomorphisms, Ergod. Th. & Dynam. Sys., 15 (1995), 161-174. doi: 10.1017/S0143385700008294.  Google Scholar

[20]

L. A. B. San Martin and L. Seco, Morse and Lyapunov spectra and dynamics on flag bundles, Ergod. Th. & Dynam. Sys., 30 (2010), 893-922. doi: 10.1017/S0143385709000285.  Google Scholar

[21]

E. D. Sontag, Finite-dimensional open-loop control generators for nonlinear systems, International J. Control, 47 (1988), 537-556. doi: 10.1080/00207178808906030.  Google Scholar

[22]

E. D. Sontag, Mathematical Control Theory. Deterministic Finite-Dimensional Systems, $2^{nd}$ edition, Texts in Applied Mathematics, 6. Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0577-7.  Google Scholar

[23]

Y. Yomdin, Volume growth and entropy, Israel J. Math., 57 (1987), 285-300. doi: 10.1007/BF02766215.  Google Scholar

[24]

L.-S. Young, Large deviations in dynamical systems, Trans. Amer. Math. Soc., 318 (1990), 525-543. doi: 10.2307/2001318.  Google Scholar

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