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April  2011, 30(1): 169-186. doi: 10.3934/dcds.2011.30.169

Upper and lower estimates for invariance entropy

1. 

Institut für Mathematik, Universität Augsburg, 86135 Augsburg, Germany

Received  November 2009 Revised  November 2010 Published  February 2011

Invariance entropy for continuous-time control systems measures how often open-loop control functions have to be updated in order to render a subset of the state space invariant. In the present paper, we derive upper and lower bounds for the invariance entropy of control systems on smooth manifolds, using differential-geometric tools. As an example, we compute these bounds explicitly for projected bilinear control systems on the unit sphere. Moreover, we derive a formula for the invariance entropy of a control set for one-dimensional control-affine systems with a single control vector field.
Citation: Christoph Kawan. Upper and lower estimates for invariance entropy. Discrete & Continuous Dynamical Systems, 2011, 30 (1) : 169-186. doi: 10.3934/dcds.2011.30.169
References:
[1]

V. A. Boichenko and G. A. Leonov, The direct Lyapunov method in estimates for topological entropy, J. Math. Sci., 91 (1998), 3370-3379 (English. Russian original); translation from Zap. Nauchn. Semin. POMI, 231 (1995), 62-75.  Google Scholar

[2]

V. A. Boichenko, G. A. Leonov and V. Reitmann, "Dimension Theory for Ordinary Differential Equations," Teubner Texts in Mathematics, Wiesbaden, 2005.  Google Scholar

[3]

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

[4]

F. Colonius and W. Kliemann, "The Dynamics of Control," Birkhäuser-Verlag, Boston, 2000.  Google Scholar

[5]

S. Gallot, D. Hulin and J. Lafontaine, "Riemannian Geometry," Springer-Verlag, Berlin, 1987.  Google Scholar

[6]

K. A. Grasse and H. J. Sussmann, Global controllability by nice controls, in "Nonlinear Controllability and Optimal Control" (H. J. Sussmann ed.), Monographs and Textbooks in Pure and Applied Mathematics 133, Marcel Dekker Inc., New York (1990), 33-79.  Google Scholar

[7]

F. Ito, An estimate from above for the entropy and the topological entropy of a $C^1$-diffeomorphism, Proc. Japan Acad., 46 (1970), 226-230. doi: 10.3792/pja/1195520395.  Google Scholar

[8]

C. Kawan, "Invariance Entropy for Control Systems," Ph.D thesis, Institut für Mathematik, Universität Augsburg, 2009, available at http://opus.bibliothek.uni-augsburg.de/volltexte/2010/1506/. Google Scholar

[9]

C. Kawan, Invariance entropy of control sets,, to appear in SIAM J. Control Optim., ().   Google Scholar

[10]

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

[11]

A. Noack, "Dimension and Entropy Estimates and Stability Investigations for Nonlinear Systems on Manifolds (Dimensions- und Entropieabschätzungen sowie Stabilitätsuntersuchungen für nichtlineare Systeme auf Mannigfaltigkeiten)," Ph.D thesis (German), Universität Dresden, 1998. Google Scholar

show all references

References:
[1]

V. A. Boichenko and G. A. Leonov, The direct Lyapunov method in estimates for topological entropy, J. Math. Sci., 91 (1998), 3370-3379 (English. Russian original); translation from Zap. Nauchn. Semin. POMI, 231 (1995), 62-75.  Google Scholar

[2]

V. A. Boichenko, G. A. Leonov and V. Reitmann, "Dimension Theory for Ordinary Differential Equations," Teubner Texts in Mathematics, Wiesbaden, 2005.  Google Scholar

[3]

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

[4]

F. Colonius and W. Kliemann, "The Dynamics of Control," Birkhäuser-Verlag, Boston, 2000.  Google Scholar

[5]

S. Gallot, D. Hulin and J. Lafontaine, "Riemannian Geometry," Springer-Verlag, Berlin, 1987.  Google Scholar

[6]

K. A. Grasse and H. J. Sussmann, Global controllability by nice controls, in "Nonlinear Controllability and Optimal Control" (H. J. Sussmann ed.), Monographs and Textbooks in Pure and Applied Mathematics 133, Marcel Dekker Inc., New York (1990), 33-79.  Google Scholar

[7]

F. Ito, An estimate from above for the entropy and the topological entropy of a $C^1$-diffeomorphism, Proc. Japan Acad., 46 (1970), 226-230. doi: 10.3792/pja/1195520395.  Google Scholar

[8]

C. Kawan, "Invariance Entropy for Control Systems," Ph.D thesis, Institut für Mathematik, Universität Augsburg, 2009, available at http://opus.bibliothek.uni-augsburg.de/volltexte/2010/1506/. Google Scholar

[9]

C. Kawan, Invariance entropy of control sets,, to appear in SIAM J. Control Optim., ().   Google Scholar

[10]

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

[11]

A. Noack, "Dimension and Entropy Estimates and Stability Investigations for Nonlinear Systems on Manifolds (Dimensions- und Entropieabschätzungen sowie Stabilitätsuntersuchungen für nichtlineare Systeme auf Mannigfaltigkeiten)," Ph.D thesis (German), Universität Dresden, 1998. Google Scholar

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