American Institute of Mathematical Sciences

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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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[9]

L. Grüne, Subdivision techniques for the computation of domains of attraction and reachable sets, in NOLCOS 2001, (2001), 762-767. Google Scholar

[10]

W. Hahn, Stability of Motion, Springer-Verlag, Berlin, 1967.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[14]

P. Koltai, A stochastic approach for computing the domain of attraction without trajectory simulation, Disc. Cont. Dynam. Sys., Supplement, II (2011), 854-863.  Google Scholar

[15]

H. J. Kushner, Probability Methods for Approximations in Stochastic Control and for Elliptic Equations, Academic Press, New York, 1977.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[18]

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002. doi: 10.1017/CBO9780511791253.  Google Scholar

[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, ().   Google Scholar

[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.  Google Scholar

[21]

J. R. Norris, Markov Chains, Cambridge Univ. Press, 1998.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

V. I. Zubov, Methods of A.M. Lyapunov and Their Application, P. Noordhoff, Groningen, 1964.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[9]

L. Grüne, Subdivision techniques for the computation of domains of attraction and reachable sets, in NOLCOS 2001, (2001), 762-767. Google Scholar

[10]

W. Hahn, Stability of Motion, Springer-Verlag, Berlin, 1967.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[14]

P. Koltai, A stochastic approach for computing the domain of attraction without trajectory simulation, Disc. Cont. Dynam. Sys., Supplement, II (2011), 854-863.  Google Scholar

[15]

H. J. Kushner, Probability Methods for Approximations in Stochastic Control and for Elliptic Equations, Academic Press, New York, 1977.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[18]

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002. doi: 10.1017/CBO9780511791253.  Google Scholar

[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, ().   Google Scholar

[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.  Google Scholar

[21]

J. R. Norris, Markov Chains, Cambridge Univ. Press, 1998.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

V. I. Zubov, Methods of A.M. Lyapunov and Their Application, P. Noordhoff, Groningen, 1964.  Google Scholar

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