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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 - A, 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.   Google Scholar

[2]

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

[3]

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

[4]

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

[5]

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

[6]

K. A. Grasse and H. J. Sussmann, Global controllability by nice controls,, in, 133 (1990), 33.   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.  doi: 10.3792/pja/1195520395.  Google Scholar

[8]

C. Kawan, "Invariance Entropy for Control Systems,", Ph.D thesis, (2009).   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.  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), (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.   Google Scholar

[2]

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

[3]

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

[4]

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

[5]

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

[6]

K. A. Grasse and H. J. Sussmann, Global controllability by nice controls,, in, 133 (1990), 33.   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.  doi: 10.3792/pja/1195520395.  Google Scholar

[8]

C. Kawan, "Invariance Entropy for Control Systems,", Ph.D thesis, (2009).   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.  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), (1998).   Google Scholar

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