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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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 |
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 and 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. |
| [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. |
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. |
| [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. |
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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