# American Institute of Mathematical Sciences

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

 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.
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

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##### 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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