Invariance entropy of hyperbolic control sets

Previous Article
The general recombination equation in continuous time and its solution
DCDS Home
This Issue
Next Article
Linearization of solution operators for state-dependent delay equations: A simple example

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
Related Papers

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

References:

[1]	E. Akin, Simplicial dynamical systems,, Mem. Amer. Math. Soc., 140* (1999). doi: 10.1090/memo/0667. Google Scholar*
[2]	V. A. Boichenko, G. A. Leonov and V. Reitmann, Dimension Theory for Ordinary Differential Equations,, Teubner, (2005). doi: 10.1007/978-3-322-80055-8. Google Scholar
[3]	R. Bowen, Topological entropy and axiom $A$,, Global Analysis, 14 (1970), 23. Google Scholar
[4]	R. Bowen, Periodic orbits for hyperbolic flows,, Amer. J. Math., 94 (1972), 1. 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. 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. doi: 10.1137/080713902. Google Scholar
[7]	F. Colonius and W. Kliemann, The Dynamics of Control,, Birkhäuser, (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 (2014). Google Scholar
[9]	J.-M. Coron, Linearized control systems and applications to smooth stabilization,, SIAM J. Control Optim., 32 (1994), 358. doi: 10.1137/S0363012992226867. Google Scholar
[10]	A. Da Silva and C. Kawan, Hyperbolic chain control sets on flag manifolds, preprint,, submitted (Nov. 2014), (2014). Google Scholar
[11]	M. F. Demers and L.-S. Young, Escape rates and conditionally invariant measures,, Nonlinearity, 19 (2006), 377. 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. doi: 10.1137/100783340. Google Scholar
[13]	C. Kawan, Invariance Entropy for Deterministic Control Systems - An Introduction,, Lecture Notes in Mathematics, 2089 (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. 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. doi: 10.1017/S0143385798100391. Google Scholar
[16]	P.-D. Liu, Random perturbations of axiom $A$ basic sets,, J. Stat. Phys., 90 (1998), 467. 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. doi: 10.1109/TAC.2004.834105. Google Scholar
[18]	H. Nijmeijer and A. van der Schaft, Nonlinear Dynamical Control Systems,, Springer-Verlag, (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. 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. 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. doi: 10.1080/00207178808906030. Google Scholar
[22]	E. D. Sontag, Mathematical Control Theory. Deterministic Finite-Dimensional Systems,, $2^{nd}$ edition, 6 (1998). doi: 10.1007/978-1-4612-0577-7. Google Scholar
[23]	Y. Yomdin, Volume growth and entropy,, Israel J. Math., 57 (1987), 285. doi: 10.1007/BF02766215. Google Scholar
[24]	L.-S. Young, Large deviations in dynamical systems,, Trans. Amer. Math. Soc., 318 (1990), 525. doi: 10.2307/2001318. Google Scholar

show all references

References:

[1]	E. Akin, Simplicial dynamical systems,, Mem. Amer. Math. Soc., 140* (1999). doi: 10.1090/memo/0667. Google Scholar*
[2]	V. A. Boichenko, G. A. Leonov and V. Reitmann, Dimension Theory for Ordinary Differential Equations,, Teubner, (2005). doi: 10.1007/978-3-322-80055-8. Google Scholar
[3]	R. Bowen, Topological entropy and axiom $A$,, Global Analysis, 14 (1970), 23. Google Scholar
[4]	R. Bowen, Periodic orbits for hyperbolic flows,, Amer. J. Math., 94 (1972), 1. 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. 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. doi: 10.1137/080713902. Google Scholar
[7]	F. Colonius and W. Kliemann, The Dynamics of Control,, Birkhäuser, (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 (2014). Google Scholar
[9]	J.-M. Coron, Linearized control systems and applications to smooth stabilization,, SIAM J. Control Optim., 32 (1994), 358. doi: 10.1137/S0363012992226867. Google Scholar
[10]	A. Da Silva and C. Kawan, Hyperbolic chain control sets on flag manifolds, preprint,, submitted (Nov. 2014), (2014). Google Scholar
[11]	M. F. Demers and L.-S. Young, Escape rates and conditionally invariant measures,, Nonlinearity, 19 (2006), 377. 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. doi: 10.1137/100783340. Google Scholar
[13]	C. Kawan, Invariance Entropy for Deterministic Control Systems - An Introduction,, Lecture Notes in Mathematics, 2089 (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. 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. doi: 10.1017/S0143385798100391. Google Scholar
[16]	P.-D. Liu, Random perturbations of axiom $A$ basic sets,, J. Stat. Phys., 90 (1998), 467. 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. doi: 10.1109/TAC.2004.834105. Google Scholar
[18]	H. Nijmeijer and A. van der Schaft, Nonlinear Dynamical Control Systems,, Springer-Verlag, (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. 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. 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. doi: 10.1080/00207178808906030. Google Scholar
[22]	E. D. Sontag, Mathematical Control Theory. Deterministic Finite-Dimensional Systems,, $2^{nd}$ edition, 6 (1998). doi: 10.1007/978-1-4612-0577-7. Google Scholar
[23]	Y. Yomdin, Volume growth and entropy,, Israel J. Math., 57 (1987), 285. doi: 10.1007/BF02766215. Google Scholar
[24]	L.-S. Young, Large deviations in dynamical systems,, Trans. Amer. Math. Soc., 318 (1990), 525. doi: 10.2307/2001318. Google Scholar

[1]	Victor Ayala, Adriano Da Silva, Luiz A. B. San Martin. Control systems on flag manifolds and their chain control sets. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2301-2313. doi: 10.3934/dcds.2017101
[2]	Fritz Colonius. Invariance entropy, quasi-stationary measures and control sets. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2093-2123. doi: 10.3934/dcds.2018086
[3]	Elena Goncharova, Maxim Staritsyn. On BV-extension of asymptotically constrained control-affine systems and complementarity problem for measure differential equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1061-1070. doi: 10.3934/dcdss.2018061
[4]	Roberta Fabbri, Sylvia Novo, Carmen Núñez, Rafael Obaya. Null controllable sets and reachable sets for nonautonomous linear control systems. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1069-1094. doi: 10.3934/dcdss.2016042
[5]	Shaobo Gan. A generalized shadowing lemma. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 627-632. doi: 10.3934/dcds.2002.8.627
[6]	Changzhi Wu, Kok Lay Teo, Volker Rehbock. Optimal control of piecewise affine systems with piecewise affine state feedback. Journal of Industrial & Management Optimization, 2009, 5 (4) : 737-747. doi: 10.3934/jimo.2009.5.737
[7]	Dietmar Szolnoki. Set oriented methods for computing reachable sets and control sets. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 361-382. doi: 10.3934/dcdsb.2003.3.361
[8]	Robert Baier, Matthias Gerdts, Ilaria Xausa. Approximation of reachable sets using optimal control algorithms. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 519-548. doi: 10.3934/naco.2013.3.519
[9]	Getachew K. Befekadu, Eduardo L. Pasiliao. On the hierarchical optimal control of a chain of distributed systems. Journal of Dynamics & Games, 2015, 2 (2) : 187-199. doi: 10.3934/jdg.2015.2.187
[10]	Byungik Kahng, Miguel Mendes. The characterization of maximal invariant sets of non-linear discrete-time control dynamical systems. Conference Publications, 2013, 2013 (special) : 393-406. doi: 10.3934/proc.2013.2013.393
[11]	Fritz Colonius, Alexandre J. Santana. Topological conjugacy for affine-linear flows and control systems. Communications on Pure & Applied Analysis, 2011, 10 (3) : 847-857. doi: 10.3934/cpaa.2011.10.847
[12]	Antonio Fernández, Pedro L. García. Regular discretizations in optimal control theory. Journal of Geometric Mechanics, 2013, 5 (4) : 415-432. doi: 10.3934/jgm.2013.5.415
[13]	Vadim Azhmyakov, Alex Poznyak, Omar Gonzalez. On the robust control design for a class of nonlinearly affine control systems: The attractive ellipsoid approach. Journal of Industrial & Management Optimization, 2013, 9 (3) : 579-593. doi: 10.3934/jimo.2013.9.579
[14]	Enrique Fernández-Cara, Diego A. Souza. On the control of some coupled systems of the Boussinesq kind with few controls. Mathematical Control & Related Fields, 2012, 2 (2) : 121-140. doi: 10.3934/mcrf.2012.2.121
[15]	Amadeu Delshams, Marian Gidea, Pablo Roldán. Transition map and shadowing lemma for normally hyperbolic invariant manifolds. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1089-1112. doi: 10.3934/dcds.2013.33.1089
[16]	Rasul Shafikov, Christian Wolf. Stable sets, hyperbolicity and dimension. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 403-412. doi: 10.3934/dcds.2005.12.403
[17]	François Blanchard, Wen Huang. Entropy sets, weakly mixing sets and entropy capacity. Discrete & Continuous Dynamical Systems - A, 2008, 20 (2) : 275-311. doi: 10.3934/dcds.2008.20.275
[18]	Andrew D. Lewis. Linearisation of tautological control systems. Journal of Geometric Mechanics, 2016, 8 (1) : 99-138. doi: 10.3934/jgm.2016.8.99
[19]	Peter E. Kloeden. Asymptotic invariance and the discretisation of nonautonomous forward attracting sets. Journal of Computational Dynamics, 2016, 3 (2) : 179-189. doi: 10.3934/jcd.2016009
[20]	Xufeng Guo, Gang Liao, Wenxiang Sun, Dawei Yang. On the hybrid control of metric entropy for dominated splittings. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5011-5019. doi: 10.3934/dcds.2018219

2018 Impact Factor: 1.143

American Institute of Mathematical Sciences