Invariance entropy of hyperbolic control sets

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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

Abstract
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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 and 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.
[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.
[3]	R. Bowen, Topological entropy and axiom $A$, Global Analysis, Proc. Sympos. Pure Math., 14 (1970), 23-41.
[4]	R. Bowen, Periodic orbits for hyperbolic flows, Amer. J. Math., 94 (1972), 1-30. doi: 10.2307/2373590.
[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.
[6]	F. Colonius and C. Kawan, Invariance entropy for control systems, SIAM J. Control Optim., 48 (2009), 1701-1721. doi: 10.1137/080713902.
[7]	F. Colonius and W. Kliemann, The Dynamics of Control, Birkhäuser, Boston, 2000. doi: 10.1007/978-1-4612-1350-5.
[8]	F. Colonius and W. Kliemann, Dynamical Systems and Linear Algebra, Graduate Studies in Mathematics, 158, AMS, 2014.
[9]	J.-M. Coron, Linearized control systems and applications to smooth stabilization, SIAM J. Control Optim., 32 (1994), 358-386. doi: 10.1137/S0363012992226867.
[10]	A. Da Silva and C. Kawan, Hyperbolic chain control sets on flag manifolds, preprint, submitted (Nov. 2014), arXiv:1402.5841.
[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.
[12]	C. Kawan, Invariance entropy of control sets, SIAM J. Control Optim., 49 (2011), 732-751. doi: 10.1137/100783340.
[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.
[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.
[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.
[16]	P.-D. Liu, Random perturbations of axiom $A$ basic sets, J. Stat. Phys., 90 (1998), 467-490. doi: 10.1023/A:1023280407906.
[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.
[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.
[19]	M. Qian and Z. Zhang, Ergodic theory for axiom $A$ endomorphisms, Ergod. Th. & Dynam. Sys., 15 (1995), 161-174. doi: 10.1017/S0143385700008294.
[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.
[21]	E. D. Sontag, Finite-dimensional open-loop control generators for nonlinear systems, International J. Control, 47 (1988), 537-556. doi: 10.1080/00207178808906030.
[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.
[23]	Y. Yomdin, Volume growth and entropy, Israel J. Math., 57 (1987), 285-300. doi: 10.1007/BF02766215.
[24]	L.-S. Young, Large deviations in dynamical systems, Trans. Amer. Math. Soc., 318 (1990), 525-543. doi: 10.2307/2001318.

show all references

References:

[1]	E. Akin, Simplicial dynamical systems, Mem. Amer. Math. Soc., 140 (1999), x+197 pp. doi: 10.1090/memo/0667.
[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.
[3]	R. Bowen, Topological entropy and axiom $A$, Global Analysis, Proc. Sympos. Pure Math., 14 (1970), 23-41.
[4]	R. Bowen, Periodic orbits for hyperbolic flows, Amer. J. Math., 94 (1972), 1-30. doi: 10.2307/2373590.
[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.
[6]	F. Colonius and C. Kawan, Invariance entropy for control systems, SIAM J. Control Optim., 48 (2009), 1701-1721. doi: 10.1137/080713902.
[7]	F. Colonius and W. Kliemann, The Dynamics of Control, Birkhäuser, Boston, 2000. doi: 10.1007/978-1-4612-1350-5.
[8]	F. Colonius and W. Kliemann, Dynamical Systems and Linear Algebra, Graduate Studies in Mathematics, 158, AMS, 2014.
[9]	J.-M. Coron, Linearized control systems and applications to smooth stabilization, SIAM J. Control Optim., 32 (1994), 358-386. doi: 10.1137/S0363012992226867.
[10]	A. Da Silva and C. Kawan, Hyperbolic chain control sets on flag manifolds, preprint, submitted (Nov. 2014), arXiv:1402.5841.
[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.
[12]	C. Kawan, Invariance entropy of control sets, SIAM J. Control Optim., 49 (2011), 732-751. doi: 10.1137/100783340.
[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.
[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.
[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.
[16]	P.-D. Liu, Random perturbations of axiom $A$ basic sets, J. Stat. Phys., 90 (1998), 467-490. doi: 10.1023/A:1023280407906.
[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.
[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.
[19]	M. Qian and Z. Zhang, Ergodic theory for axiom $A$ endomorphisms, Ergod. Th. & Dynam. Sys., 15 (1995), 161-174. doi: 10.1017/S0143385700008294.
[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.
[21]	E. D. Sontag, Finite-dimensional open-loop control generators for nonlinear systems, International J. Control, 47 (1988), 537-556. doi: 10.1080/00207178808906030.
[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.
[23]	Y. Yomdin, Volume growth and entropy, Israel J. Math., 57 (1987), 285-300. doi: 10.1007/BF02766215.
[24]	L.-S. Young, Large deviations in dynamical systems, Trans. Amer. Math. Soc., 318 (1990), 525-543. doi: 10.2307/2001318.

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