# American Institute of Mathematical Sciences

April 2018, 38(4): 2093-2123. doi: 10.3934/dcds.2018086

## Invariance entropy, quasi-stationary measures and control sets

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

Received  May 2017 Revised  October 2017 Published  January 2018

Fund Project: Research supported by DFG grant 124/19-2

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.

Citation: 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
##### References:
 [1] F. Albertini and E. D. Sontag, Some connections between chaotic dynamical systems and control systems, in Proc. European Control Conference, Grenoble, 1991,158-163. [2] ———, Discrete-time transitivity and accessibility: Analytic systems, SIAM J. Control Optim., 31 (1993), 1599-1622. doi: 10.1137/0331075. [3] M. Benaïm, B. Cloez and F. Panloup, Stochastic Approximation of Quasi-Stationary Distributions on Compact Spaces and Applications, arXiv: 1606.06477v2 [math. PR] 6 Dec 2016. [4] T. Bogenschütz, Entropy, pressure, and a variational principle for random dynamical systems, Random and Computational Dynamics, 1 (1992/93), 99-116. [5] P. Collett, S. Martinez and J. San Martin, Quasi-Stationary Distributions: Markov Chains, Diffusions, and Dynamical Systems, Springer-Verlag, Berlin, 2013. [6] F. Colonius, Metric invariance entropy and conditionally invariant measures Ergodic Theory and Dynamical Systems, (2016). First published online: 20 October 2016. doi: 10.1017/etds.2016.72. [7] ———, Metric Invariance Entropy and Relatively Invariant Control Sets, in Proceedings of the 55th IEEE Conference on Decision and Control (Las Vegas, December 12-14,2016), 2016. [8] F. Colonius, J.-A. Homburg and W. Kliemann, Near invariance and local transience for random diffeomorphisms, J. Difference Equations and Applications, 16 (2010), 127-141. doi: 10.1080/10236190802653646. [9] F. Colonius and R. Lettau, Relative controllability properties, IMA Journal of Mathematical Control and Information, 33 (2016), 701-722. doi: 10.1093/imamci/dnv004. [10] A. da Silva and C. Kawan, Invariance entropy of hyperbolic control sets, Discrete Cont. Dyn. Syst. A, 36 (2016), 97-136. [11] M. F. Demers, Introductory Lectures on Open Systems, given as part of the LMS-CMI Research School at Loughborough University, April 13-17,2015. [12] 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. [13] E.A. van Doorn and P. Pollett, Quasi-stationary distributions for reducible absorbing Markov chains in discrete time, Markov Processes and Related Fields, 15 (2009), 191-204. [14] B. Jakubczyk and E.D. Sontag, Controllability of nonlinear discrete time systems: A Lie algebraic approach, SIAM J. Control Optim., 28 (1990), 1-33. doi: 10.1137/0328001. [15] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, 1995. [16] C. Kawan, Invariance entropy of control sets, SIAM J. Control Optim., 49 (2011), 732-751. doi: 10.1137/100783340. [17] ———, Invariance Entropy for Deterministic Control Systems. An Introduction, vol. 2089 of Lecture Notes in Mathematics, Springer-Verlag, 2013. [18] S. Méléard and D. Villemonais, Quasi-stationary distributions and population processes, Probability Surveys, 9 (2012), 340-410. doi: 10.1214/11-PS191. [19] G. Nair, R.J. Evans, I. Mareels and W. Moran, Topological feedback entropy and nonlinear stabilization, IEEE Trans. Aut. Control, 49 (2004), 1585-1597. doi: 10.1109/TAC.2004.834105. [20] M. Patrão and L. San Martin, Semiflows on topological spaces: Chain transitivity and semigroups, J. Dyn. Diff. Equations, 19 (2007), 155-180. [21] P. Pollett, Quasi-stationary distributions: a bibliography. http://www.maths.uq.edu.au/~pkp/papers/qsds/qsds.pdf, 2015. [22] F. Rodrigues and P. Varandas, Specification and thermodynamical properties of semigroup actions, Journal Math. Phys. , 57 (2016), 052704, 27pp. [23] E. Sontag and F. Wirth, Remarks on universal nonsingular controls for discrete-time systems, Sys. Control Lett., 33 (1998), 81-88. doi: 10.1016/S0167-6911(97)00117-5. [24] M. Viana and K. Oliveira, Foundations of Ergodic Theory, Cambridge University Press, 2016. [25] P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, 1982. [26] F. Wirth, Robust Stability of Discrete-Time Systems under Time-Varying Perturbations, PhD thesis, Fachbereich Mathematik/Informatik, Universität Bremen, 1995. [27] F. Wirth, Dynamics and controllability of nonlinear discrete-time control systems, IFAC Proceedings Volumes, 31 (1998), 267-272. doi: 10.1016/S1474-6670(17)40346-6.

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##### References:
 [1] F. Albertini and E. D. Sontag, Some connections between chaotic dynamical systems and control systems, in Proc. European Control Conference, Grenoble, 1991,158-163. [2] ———, Discrete-time transitivity and accessibility: Analytic systems, SIAM J. Control Optim., 31 (1993), 1599-1622. doi: 10.1137/0331075. [3] M. Benaïm, B. Cloez and F. Panloup, Stochastic Approximation of Quasi-Stationary Distributions on Compact Spaces and Applications, arXiv: 1606.06477v2 [math. PR] 6 Dec 2016. [4] T. Bogenschütz, Entropy, pressure, and a variational principle for random dynamical systems, Random and Computational Dynamics, 1 (1992/93), 99-116. [5] P. Collett, S. Martinez and J. San Martin, Quasi-Stationary Distributions: Markov Chains, Diffusions, and Dynamical Systems, Springer-Verlag, Berlin, 2013. [6] F. Colonius, Metric invariance entropy and conditionally invariant measures Ergodic Theory and Dynamical Systems, (2016). First published online: 20 October 2016. doi: 10.1017/etds.2016.72. [7] ———, Metric Invariance Entropy and Relatively Invariant Control Sets, in Proceedings of the 55th IEEE Conference on Decision and Control (Las Vegas, December 12-14,2016), 2016. [8] F. Colonius, J.-A. Homburg and W. Kliemann, Near invariance and local transience for random diffeomorphisms, J. Difference Equations and Applications, 16 (2010), 127-141. doi: 10.1080/10236190802653646. [9] F. Colonius and R. Lettau, Relative controllability properties, IMA Journal of Mathematical Control and Information, 33 (2016), 701-722. doi: 10.1093/imamci/dnv004. [10] A. da Silva and C. Kawan, Invariance entropy of hyperbolic control sets, Discrete Cont. Dyn. Syst. A, 36 (2016), 97-136. [11] M. F. Demers, Introductory Lectures on Open Systems, given as part of the LMS-CMI Research School at Loughborough University, April 13-17,2015. [12] 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. [13] E.A. van Doorn and P. Pollett, Quasi-stationary distributions for reducible absorbing Markov chains in discrete time, Markov Processes and Related Fields, 15 (2009), 191-204. [14] B. Jakubczyk and E.D. Sontag, Controllability of nonlinear discrete time systems: A Lie algebraic approach, SIAM J. Control Optim., 28 (1990), 1-33. doi: 10.1137/0328001. [15] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, 1995. [16] C. Kawan, Invariance entropy of control sets, SIAM J. Control Optim., 49 (2011), 732-751. doi: 10.1137/100783340. [17] ———, Invariance Entropy for Deterministic Control Systems. An Introduction, vol. 2089 of Lecture Notes in Mathematics, Springer-Verlag, 2013. [18] S. Méléard and D. Villemonais, Quasi-stationary distributions and population processes, Probability Surveys, 9 (2012), 340-410. doi: 10.1214/11-PS191. [19] G. Nair, R.J. Evans, I. Mareels and W. Moran, Topological feedback entropy and nonlinear stabilization, IEEE Trans. Aut. Control, 49 (2004), 1585-1597. doi: 10.1109/TAC.2004.834105. [20] M. Patrão and L. San Martin, Semiflows on topological spaces: Chain transitivity and semigroups, J. Dyn. Diff. Equations, 19 (2007), 155-180. [21] P. Pollett, Quasi-stationary distributions: a bibliography. http://www.maths.uq.edu.au/~pkp/papers/qsds/qsds.pdf, 2015. [22] F. Rodrigues and P. Varandas, Specification and thermodynamical properties of semigroup actions, Journal Math. Phys. , 57 (2016), 052704, 27pp. [23] E. Sontag and F. Wirth, Remarks on universal nonsingular controls for discrete-time systems, Sys. Control Lett., 33 (1998), 81-88. doi: 10.1016/S0167-6911(97)00117-5. [24] M. Viana and K. Oliveira, Foundations of Ergodic Theory, Cambridge University Press, 2016. [25] P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, 1982. [26] F. Wirth, Robust Stability of Discrete-Time Systems under Time-Varying Perturbations, PhD thesis, Fachbereich Mathematik/Informatik, Universität Bremen, 1995. [27] F. Wirth, Dynamics and controllability of nonlinear discrete-time control systems, IFAC Proceedings Volumes, 31 (1998), 267-272. doi: 10.1016/S1474-6670(17)40346-6.
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 $\alpha = 0.08$)
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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