# 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.  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.  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.  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.  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.  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, 30 (1985), 747.  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.  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.   Google Scholar [10] W. Hahn, Stability of Motion,, Springer-Verlag, (1967).   Google Scholar [11] C. S. Hsu, A theory of cell-to-cell mapping dynamical systems,, SME J. appl. Mech., 47 (1980), 931.  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.  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, (2010).   Google Scholar [14] P. Koltai, A stochastic approach for computing the domain of attraction without trajectory simulation,, Disc. Cont. Dynam. Sys., II (2011), 854.   Google Scholar [15] H. J. Kushner, Probability Methods for Approximations in Stochastic Control and for Elliptic Equations,, Academic Press, (1977).   Google Scholar [16] H. J. Kushner and P. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time,, 2nd edition, (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, (1961).   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.   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.  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.  doi: 10.1239/jap/1032438377.  Google Scholar [24] V. I. Zubov, Methods of A.M. Lyapunov and Their Application,, P. Noordhoff, (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.  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.  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.  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.  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.  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, 30 (1985), 747.  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.  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.   Google Scholar [10] W. Hahn, Stability of Motion,, Springer-Verlag, (1967).   Google Scholar [11] C. S. Hsu, A theory of cell-to-cell mapping dynamical systems,, SME J. appl. Mech., 47 (1980), 931.  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.  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, (2010).   Google Scholar [14] P. Koltai, A stochastic approach for computing the domain of attraction without trajectory simulation,, Disc. Cont. Dynam. Sys., II (2011), 854.   Google Scholar [15] H. J. Kushner, Probability Methods for Approximations in Stochastic Control and for Elliptic Equations,, Academic Press, (1977).   Google Scholar [16] H. J. Kushner and P. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time,, 2nd edition, (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, (1961).   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.   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.  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.  doi: 10.1239/jap/1032438377.  Google Scholar [24] V. I. Zubov, Methods of A.M. Lyapunov and Their Application,, P. Noordhoff, (1964).   Google Scholar
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