Optimizing the stable behavior of parameter-dependent dynamical systems --- maximal domains of attraction, minimal absorption times

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June 2014, 1(2): 339-356. doi: 10.3934/jcd.2014.1.339

Optimizing the stable behavior of parameter-dependent dynamical systems --- maximal domains of attraction, minimal absorption times

Péter Koltai ^1, and Alexander Volf ^2,

1.	Freie Universität Berlin, Department of Mathematics and Computer Science, Arnimallee 6, 14195 Berlin, Germany
2.	Klinikum rechts der Isar der Technischen Universität München, Dept. of Plastic and Reconstructive Surgery, Ismaninger Straße 22, München, Germany

Received October 2011 Revised May 2012 Published December 2014

Abstract
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We propose a method for approximating solutions to optimization problems involving the global stability properties of parameter-dependent continuous-time autonomous dynamical systems. The method relies on an approximation of the infinite-state deterministic system by a finite-state non-deterministic one --- a Markov jump process. The key properties of the method are that it does not use any trajectory simulation, and that the parameters and objective function are in a simple (and except for a system of linear equations) explicit relationship.

Keywords: absorption time, stability design, parameterdependent system, Domain of attraction, Markov jump process..

Mathematics Subject Classification: Primary: 34D23, 31C20; Secondary: 37C75, 60J27, 37M9.

Citation: Péter Koltai, Alexander Volf. Optimizing the stable behavior of parameter-dependent dynamical systems --- maximal domains of attraction, minimal absorption times. Journal of Computational Dynamics, 2014, 1 (2) : 339-356. doi: 10.3934/jcd.2014.1.339

References:

[1]	E. J. Davison and E. M. Kurak, A computational method for determining quadratic Lyapunov functions for non-linear systems, Automatica, 7 (1971), 627-636, URL http://www.sciencedirect.com/science/article/pii/0005109871900276. doi: 10.1016/0005-1098(71)90027-6.
[2]	M. Dellnitz and A. Hohmann, A subdivision algorithm for the computation of unstable manifolds and global attractors, Numerische Mathematik, 75 (1997), 293-317. doi: 10.1007/s002110050240.
[3]	M. Dellnitz and O. Junge, An adaptive subdivision technique for the approximation of attractors and invariant measures, Comput. Visual. Sci., 1 (1998), 63-68. doi: 10.1007/s007910050006.
[4]	H. Flashner and R. S. Guttalu, A computational approach for studying domains of attraction for non-linear systems, Int. J. Non-Linear Mech., 23 (1988), 279-295. doi: 10.1016/0020-7462(88)90026-1.
[5]	G. Froyland, O. Junge and P. Koltai, Estimating long term behavior of flows without trajectory integration: the infinitesimal generator approach, SIAM J. Numer. Anal., 51 (2013), 223-247. doi: 10.1137/110819986.
[6]	R. Genesio, M. Tartaglia and A. Vicino, On the estimation of asymptotic stability regions: State of the art and new proposals, Automatic Control, IEEE Transactions on, 30 (1985), 747-755. doi: 10.1109/TAC.1985.1104057.
[7]	P. Giesl, On the determination of the basin of attraction of periodic orbits in three- and higher-dimensional systems, J. Math. Anal. Appl., 354 (2009), 606-618. doi: 10.1016/j.jmaa.2009.01.027.
[8]	S. Goldschmidt, N. Neumann and J. Wallaschek, On the application of set-oriented numerical methods in the analysis of railway vehicle dynamics, in ECCOMAS 2004, 2004.
[9]	L. Grüne, Subdivision techniques for the computation of domains of attraction and reachable sets, in NOLCOS 2001, (2001), 762-767.
[10]	W. Hahn, Stability of Motion, Springer-Verlag, Berlin, 1967.
[11]	C. S. Hsu, A theory of cell-to-cell mapping dynamical systems, SME J. appl. Mech., 47 (1980), 931-939. doi: 10.1115/1.3153816.
[12]	C. S. Hsu and R. S. Guttalu, An unravelling algorithm for global analysis of dynamical systems: an application of cell-to-cell mappings, ASME J. appl. Mech., 47 (1980), 940-948. doi: 10.1115/1.3153817.
[13]	P. Koltai, Efficient Approximation Methods for the Global Long-Term Behavior of Dynamical Systems - Theory, Algorithms and Examples, PhD thesis, Technische Universität München, 2010.
[14]	P. Koltai, A stochastic approach for computing the domain of attraction without trajectory simulation, Disc. Cont. Dynam. Sys., Supplement, II (2011), 854-863.
[15]	H. J. Kushner, Probability Methods for Approximations in Stochastic Control and for Elliptic Equations, Academic Press, New York, 1977.
[16]	H. J. Kushner and P. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time, 2nd edition, Springer-Verlag, New York, 1992. doi: 10.1007/978-1-4684-0441-8.
[17]	J. P. LaSalle and S. Lefschetz, Stability by Liapunov's Direct Method with Applications, Mathematics in science and engineering, Academic Press, 1961, URL http://books.google.de/books?id=UsU-AAAAIAAJ.
[18]	R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002. doi: 10.1017/CBO9780511791253.
[19]	D.-C. Liaw and C.-H. Lee, An approach to estimate domain of attraction for nonlinear control systems, Proceedings of the First International Conference on Innovative Computing, Information and Control.
[20]	Y. Nesterov, A method for unconstrained convex minimization problem with the rate of convergence $o(1/k^2)$, Doklady AN SSSR (translated as Soviet Math. Docl.), 269 (1983), 543-547.
[21]	J. R. Norris, Markov Chains, Cambridge Univ. Press, 1998.
[22]	D. N. Shields and C. Storey, The behaviour of optimal Lyapunov functions, International Journal of Control, 21 (1975), 561-573, URL http://www.tandfonline.com/doi/abs/10.1080/00207177508922012. doi: 10.1080/00207177508922012.
[23]	D. M. Walker, The expected time until absorption when absorption is not certain, J. Appl. Prob., 35 (1998), 812-823. doi: 10.1239/jap/1032438377.
[24]	V. I. Zubov, Methods of A.M. Lyapunov and Their Application, P. Noordhoff, Groningen, 1964.

show all references

References:

[1]	E. J. Davison and E. M. Kurak, A computational method for determining quadratic Lyapunov functions for non-linear systems, Automatica, 7 (1971), 627-636, URL http://www.sciencedirect.com/science/article/pii/0005109871900276. doi: 10.1016/0005-1098(71)90027-6.
[2]	M. Dellnitz and A. Hohmann, A subdivision algorithm for the computation of unstable manifolds and global attractors, Numerische Mathematik, 75 (1997), 293-317. doi: 10.1007/s002110050240.
[3]	M. Dellnitz and O. Junge, An adaptive subdivision technique for the approximation of attractors and invariant measures, Comput. Visual. Sci., 1 (1998), 63-68. doi: 10.1007/s007910050006.
[4]	H. Flashner and R. S. Guttalu, A computational approach for studying domains of attraction for non-linear systems, Int. J. Non-Linear Mech., 23 (1988), 279-295. doi: 10.1016/0020-7462(88)90026-1.
[5]	G. Froyland, O. Junge and P. Koltai, Estimating long term behavior of flows without trajectory integration: the infinitesimal generator approach, SIAM J. Numer. Anal., 51 (2013), 223-247. doi: 10.1137/110819986.
[6]	R. Genesio, M. Tartaglia and A. Vicino, On the estimation of asymptotic stability regions: State of the art and new proposals, Automatic Control, IEEE Transactions on, 30 (1985), 747-755. doi: 10.1109/TAC.1985.1104057.
[7]	P. Giesl, On the determination of the basin of attraction of periodic orbits in three- and higher-dimensional systems, J. Math. Anal. Appl., 354 (2009), 606-618. doi: 10.1016/j.jmaa.2009.01.027.
[8]	S. Goldschmidt, N. Neumann and J. Wallaschek, On the application of set-oriented numerical methods in the analysis of railway vehicle dynamics, in ECCOMAS 2004, 2004.
[9]	L. Grüne, Subdivision techniques for the computation of domains of attraction and reachable sets, in NOLCOS 2001, (2001), 762-767.
[10]	W. Hahn, Stability of Motion, Springer-Verlag, Berlin, 1967.
[11]	C. S. Hsu, A theory of cell-to-cell mapping dynamical systems, SME J. appl. Mech., 47 (1980), 931-939. doi: 10.1115/1.3153816.
[12]	C. S. Hsu and R. S. Guttalu, An unravelling algorithm for global analysis of dynamical systems: an application of cell-to-cell mappings, ASME J. appl. Mech., 47 (1980), 940-948. doi: 10.1115/1.3153817.
[13]	P. Koltai, Efficient Approximation Methods for the Global Long-Term Behavior of Dynamical Systems - Theory, Algorithms and Examples, PhD thesis, Technische Universität München, 2010.
[14]	P. Koltai, A stochastic approach for computing the domain of attraction without trajectory simulation, Disc. Cont. Dynam. Sys., Supplement, II (2011), 854-863.
[15]	H. J. Kushner, Probability Methods for Approximations in Stochastic Control and for Elliptic Equations, Academic Press, New York, 1977.
[16]	H. J. Kushner and P. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time, 2nd edition, Springer-Verlag, New York, 1992. doi: 10.1007/978-1-4684-0441-8.
[17]	J. P. LaSalle and S. Lefschetz, Stability by Liapunov's Direct Method with Applications, Mathematics in science and engineering, Academic Press, 1961, URL http://books.google.de/books?id=UsU-AAAAIAAJ.
[18]	R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002. doi: 10.1017/CBO9780511791253.
[19]	D.-C. Liaw and C.-H. Lee, An approach to estimate domain of attraction for nonlinear control systems, Proceedings of the First International Conference on Innovative Computing, Information and Control.
[20]	Y. Nesterov, A method for unconstrained convex minimization problem with the rate of convergence $o(1/k^2)$, Doklady AN SSSR (translated as Soviet Math. Docl.), 269 (1983), 543-547.
[21]	J. R. Norris, Markov Chains, Cambridge Univ. Press, 1998.
[22]	D. N. Shields and C. Storey, The behaviour of optimal Lyapunov functions, International Journal of Control, 21 (1975), 561-573, URL http://www.tandfonline.com/doi/abs/10.1080/00207177508922012. doi: 10.1080/00207177508922012.
[23]	D. M. Walker, The expected time until absorption when absorption is not certain, J. Appl. Prob., 35 (1998), 812-823. doi: 10.1239/jap/1032438377.
[24]	V. I. Zubov, Methods of A.M. Lyapunov and Their Application, P. Noordhoff, Groningen, 1964.

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