-
Previous Article
Linearization of solution operators for state-dependent delay equations: A simple example
- DCDS Home
- This Issue
-
Next Article
The general recombination equation in continuous time and its solution
Invariance entropy of hyperbolic control sets
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 |
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. |
[1] |
Victor Ayala, Adriano Da Silva, Luiz A. B. San Martin. Control systems on flag manifolds and their chain control sets. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2301-2313. doi: 10.3934/dcds.2017101 |
[2] |
Fritz Colonius. Invariance entropy, quasi-stationary measures and control sets. Discrete and Continuous Dynamical Systems, 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 and 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 and Continuous Dynamical Systems - S, 2016, 9 (4) : 1069-1094. doi: 10.3934/dcdss.2016042 |
[5] |
Shaobo Gan. A generalized shadowing lemma. Discrete and Continuous Dynamical Systems, 2002, 8 (3) : 627-632. doi: 10.3934/dcds.2002.8.627 |
[6] |
Dietmar Szolnoki. Set oriented methods for computing reachable sets and control sets. Discrete and Continuous Dynamical Systems - B, 2003, 3 (3) : 361-382. doi: 10.3934/dcdsb.2003.3.361 |
[7] |
Robert Baier, Matthias Gerdts, Ilaria Xausa. Approximation of reachable sets using optimal control algorithms. Numerical Algebra, Control and Optimization, 2013, 3 (3) : 519-548. doi: 10.3934/naco.2013.3.519 |
[8] |
Changzhi Wu, Kok Lay Teo, Volker Rehbock. Optimal control of piecewise affine systems with piecewise affine state feedback. Journal of Industrial and Management Optimization, 2009, 5 (4) : 737-747. doi: 10.3934/jimo.2009.5.737 |
[9] |
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 |
[10] |
Getachew K. Befekadu, Eduardo L. Pasiliao. On the hierarchical optimal control of a chain of distributed systems. Journal of Dynamics and Games, 2015, 2 (2) : 187-199. doi: 10.3934/jdg.2015.2.187 |
[11] |
Fritz Colonius, Alexandre J. Santana. Topological conjugacy for affine-linear flows and control systems. Communications on Pure and Applied Analysis, 2011, 10 (3) : 847-857. doi: 10.3934/cpaa.2011.10.847 |
[12] |
Rasul Shafikov, Christian Wolf. Stable sets, hyperbolicity and dimension. Discrete and Continuous Dynamical Systems, 2005, 12 (3) : 403-412. doi: 10.3934/dcds.2005.12.403 |
[13] |
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 |
[14] |
Enrique Fernández-Cara, Diego A. Souza. On the control of some coupled systems of the Boussinesq kind with few controls. Mathematical Control and Related Fields, 2012, 2 (2) : 121-140. doi: 10.3934/mcrf.2012.2.121 |
[15] |
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 and Management Optimization, 2013, 9 (3) : 579-593. doi: 10.3934/jimo.2013.9.579 |
[16] |
Amadeu Delshams, Marian Gidea, Pablo Roldán. Transition map and shadowing lemma for normally hyperbolic invariant manifolds. Discrete and Continuous Dynamical Systems, 2013, 33 (3) : 1089-1112. doi: 10.3934/dcds.2013.33.1089 |
[17] |
Marc Puche, Timo Reis, Felix L. Schwenninger. Funnel control for boundary control systems. Evolution Equations and Control Theory, 2021, 10 (3) : 519-544. doi: 10.3934/eect.2020079 |
[18] |
François Blanchard, Wen Huang. Entropy sets, weakly mixing sets and entropy capacity. Discrete and Continuous Dynamical Systems, 2008, 20 (2) : 275-311. doi: 10.3934/dcds.2008.20.275 |
[19] |
Shuhei Hayashi. Erratum and addendum to "A forward Ergodic Closing Lemma and the Entropy Conjecture for nonsingular endomorphisms away from tangencies" (Volume 40, Number 4, 2020, 2285-2313). Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2433-2437. doi: 10.3934/dcds.2021196 |
[20] |
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 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]