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

Invariance entropy, quasi-stationary measures and control sets

Fritz Colonius

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.

Keywords: Invariance entropy, quasi-stationary measures, control sets, coder-controllers, transitivity set.
Mathematics Subject Classification: Primary: 93C41, 37A35; Secondary: 94A17.
Citation: Fritz Colonius. Invariance entropy, quasi-stationary measures and control sets. Discrete & Continuous Dynamical Systems, 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. Google Scholar

[2]

———, Discrete-time transitivity and accessibility: Analytic systems, SIAM J. Control Optim., 31 (1993), 1599-1622. doi: 10.1137/0331075.  Google Scholar

[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. Google Scholar

[4]

T. Bogenschütz, Entropy, pressure, and a variational principle for random dynamical systems, Random and Computational Dynamics, 1 (1992/93), 99-116.   Google Scholar

[5]

P. Collett, S. Martinez and J. San Martin, Quasi-Stationary Distributions: Markov Chains, Diffusions, and Dynamical Systems, Springer-Verlag, Berlin, 2013.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

A. da Silva and C. Kawan, Invariance entropy of hyperbolic control sets, Discrete Cont. Dyn. Syst. A, 36 (2016), 97-136.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[15]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, 1995.  Google Scholar

[16]

C. Kawan, Invariance entropy of control sets, SIAM J. Control Optim., 49 (2011), 732-751.  doi: 10.1137/100783340.  Google Scholar

[17]

———, Invariance Entropy for Deterministic Control Systems. An Introduction, vol. 2089 of Lecture Notes in Mathematics, Springer-Verlag, 2013.  Google Scholar

[18]

S. Méléard and D. Villemonais, Quasi-stationary distributions and population processes, Probability Surveys, 9 (2012), 340-410.  doi: 10.1214/11-PS191.  Google Scholar

[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.  Google Scholar

[20]

M. Patrão and L. San Martin, Semiflows on topological spaces: Chain transitivity and semigroups, J. Dyn. Diff. Equations, 19 (2007), 155-180.   Google Scholar

[21]

P. Pollett, Quasi-stationary distributions: a bibliography. http://www.maths.uq.edu.au/~pkp/papers/qsds/qsds.pdf, 2015. Google Scholar

[22]

F. Rodrigues and P. Varandas, Specification and thermodynamical properties of semigroup actions, Journal Math. Phys. , 57 (2016), 052704, 27pp.  Google Scholar

[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.  Google Scholar

[24]

M. Viana and K. Oliveira, Foundations of Ergodic Theory, Cambridge University Press, 2016.  Google Scholar

[25]

P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, 1982.  Google Scholar

[26]

F. Wirth, Robust Stability of Discrete-Time Systems under Time-Varying Perturbations, PhD thesis, Fachbereich Mathematik/Informatik, Universität Bremen, 1995. Google Scholar

[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.  Google Scholar

show all references

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. Google Scholar

[2]

———, Discrete-time transitivity and accessibility: Analytic systems, SIAM J. Control Optim., 31 (1993), 1599-1622. doi: 10.1137/0331075.  Google Scholar

[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. Google Scholar

[4]

T. Bogenschütz, Entropy, pressure, and a variational principle for random dynamical systems, Random and Computational Dynamics, 1 (1992/93), 99-116.   Google Scholar

[5]

P. Collett, S. Martinez and J. San Martin, Quasi-Stationary Distributions: Markov Chains, Diffusions, and Dynamical Systems, Springer-Verlag, Berlin, 2013.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

A. da Silva and C. Kawan, Invariance entropy of hyperbolic control sets, Discrete Cont. Dyn. Syst. A, 36 (2016), 97-136.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[15]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, 1995.  Google Scholar

[16]

C. Kawan, Invariance entropy of control sets, SIAM J. Control Optim., 49 (2011), 732-751.  doi: 10.1137/100783340.  Google Scholar

[17]

———, Invariance Entropy for Deterministic Control Systems. An Introduction, vol. 2089 of Lecture Notes in Mathematics, Springer-Verlag, 2013.  Google Scholar

[18]

S. Méléard and D. Villemonais, Quasi-stationary distributions and population processes, Probability Surveys, 9 (2012), 340-410.  doi: 10.1214/11-PS191.  Google Scholar

[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.  Google Scholar

[20]

M. Patrão and L. San Martin, Semiflows on topological spaces: Chain transitivity and semigroups, J. Dyn. Diff. Equations, 19 (2007), 155-180.   Google Scholar

[21]

P. Pollett, Quasi-stationary distributions: a bibliography. http://www.maths.uq.edu.au/~pkp/papers/qsds/qsds.pdf, 2015. Google Scholar

[22]

F. Rodrigues and P. Varandas, Specification and thermodynamical properties of semigroup actions, Journal Math. Phys. , 57 (2016), 052704, 27pp.  Google Scholar

[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.  Google Scholar

[24]

M. Viana and K. Oliveira, Foundations of Ergodic Theory, Cambridge University Press, 2016.  Google Scholar

[25]

P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, 1982.  Google Scholar

[26]

F. Wirth, Robust Stability of Discrete-Time Systems under Time-Varying Perturbations, PhD thesis, Fachbereich Mathematik/Informatik, Universität Bremen, 1995. Google Scholar

[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.  Google Scholar

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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