Invariance entropy, quasi-stationary measures and control sets

Fritz Colonius

doi:10.3934/dcds.2018086

Article Contents

2018, Volume 38, Issue 4: 2093-2123. Doi: 10.3934/dcds.2018086

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Invariance entropy, quasi-stationary measures and control sets

Fritz Colonius

Institut für Mathematik, Universität Augsburg, Universitätsstrasse 14,86159 Augsburg, Germany

Received: April 30, 2017

Revised: September 30, 2017

Published: June 2018

Research supported by DFG grant 124/19-2

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Abstract

For control systems in discrete time, this paper discusses measure-theoretic invariance entropy for a subset Q of the state space with respect to a quasi-stationary measure obtained by endowing the control range with a probability measure. The main results show that this entropy is invariant under measurable transformations and that it is already determined by certain subsets of Q which are characterized by controllability properties.

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Mathematics Subject Classification: Primary: 93C41, 37A35; Secondary: 94A17.

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Figure . Extremal graphs for (24) and the set $[d(\alpha),0.5]$ in $Q = [0.2,0.5\dot{]}$ (here $A = 0.05,\sigma = 0.1$ and $\alpha = 0.08$)

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Figure . Extremal graphs for (44) and the $W$-control sets $D_1(\alpha) = [a(\alpha),b(\alpha))$ and $D_2(\alpha) = [d(\alpha),0.7)$ in $Q = [0.1,0.7\dot {]}$ (here $A = 0.05,\sigma = 0.1$ and $\alpha = 0.08$)

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