American Institute of Mathematical Sciences

Advanced
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

[1]

Liqiang Jin, Yanqing Liu, Yanyan Yin, Kok Lay Teo, Fei Liu. Design of probabilistic $ l_2-l_\infty $ filter for uncertain Markov jump systems with partial information of the transition probabilities. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021070
[2]

Yueyuan Zhang, Yanyan Yin, Fei Liu. Robust observer-based control for discrete-time semi-Markov jump systems with actuator saturation. Journal of Industrial & Management Optimization, 2021, 17 (6) : 3013-3026. doi: 10.3934/jimo.2020105
[3]

Michael C. Fu, Bingqing Li, Rongwen Wu, Tianqi Zhang. Option pricing under a discrete-time Markov switching stochastic volatility with co-jump model. Frontiers of Mathematical Finance, , () : -. doi: 10.3934/fmf.2021005
[4]

Xian Chen, Zhi-Ming Ma. A transformation of Markov jump processes and applications in genetic study. Discrete & Continuous Dynamical Systems, 2014, 34 (12) : 5061-5084. doi: 10.3934/dcds.2014.34.5061
[5]

Michel Pierre, Grégory Vial. Best design for a fastest cells selecting process. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 223-237. doi: 10.3934/dcdss.2011.4.223
[6]

Charles L. Epstein, Leslie Greengard, Thomas Hagstrom. On the stability of time-domain integral equations for acoustic wave propagation. Discrete & Continuous Dynamical Systems, 2016, 36 (8) : 4367-4382. doi: 10.3934/dcds.2016.36.4367
[7]

Péter Koltai. A stochastic approach for computing the domain of attraction without trajectory simulation. Conference Publications, 2011, 2011 (Special) : 854-863. doi: 10.3934/proc.2011.2011.854
[8]

Benoît Perthame, P. E. Souganidis. Front propagation for a jump process model arising in spacial ecology. Discrete & Continuous Dynamical Systems, 2005, 13 (5) : 1235-1246. doi: 10.3934/dcds.2005.13.1235
[9]

Yanqing Liu, Yanyan Yin, Kok Lay Teo, Song Wang, Fei Liu. Probabilistic control of Markov jump systems by scenario optimization approach. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1447-1453. doi: 10.3934/jimo.2018103
[10]

Fágner D. Araruna, Flank D. M. Bezerra, Milton L. Oliveira. Rate of attraction for a semilinear thermoelastic system with variable coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3211-3226. doi: 10.3934/dcdsb.2018316
[11]

Daoyi Xu, Yumei Huang, Zhiguo Yang. Existence theorems for periodic Markov process and stochastic functional differential equations. Discrete & Continuous Dynamical Systems, 2009, 24 (3) : 1005-1023. doi: 10.3934/dcds.2009.24.1005
[12]

Joana Terra, Noemi Wolanski. Large time behavior for a nonlocal diffusion equation with absorption and bounded initial data. Discrete & Continuous Dynamical Systems, 2011, 31 (2) : 581-605. doi: 10.3934/dcds.2011.31.581
[13]

Pavel Chigansky, Fima C. Klebaner. The Euler-Maruyama approximation for the absorption time of the CEV diffusion. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1455-1471. doi: 10.3934/dcdsb.2012.17.1455
[14]

Igor Pažanin, Marcone C. Pereira. On the nonlinear convection-diffusion-reaction problem in a thin domain with a weak boundary absorption. Communications on Pure & Applied Analysis, 2018, 17 (2) : 579-592. doi: 10.3934/cpaa.2018031
[15]

Ahmed Bchatnia, Aissa Guesmia. Well-posedness and asymptotic stability for the Lamé system with infinite memories in a bounded domain. Mathematical Control & Related Fields, 2014, 4 (4) : 451-463. doi: 10.3934/mcrf.2014.4.451
[16]

Jamal Mrazgua, El Houssaine Tissir, Mohamed Ouahi. Frequency domain $ H_{\infty} $ control design for active suspension systems. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021036
[17]

Weike Wang, Yucheng Wang. Global existence and large time behavior for the chemotaxis–shallow water system in a bounded domain. Discrete & Continuous Dynamical Systems, 2020, 40 (11) : 6379-6409. doi: 10.3934/dcds.2020284
[18]

Sumon Sarkar, Bibhas C. Giri. Optimal lot-sizing policy for a failure prone production system with investment in process quality improvement and lead time variance reduction. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021048
[19]

Isabelle Kuhwald, Ilya Pavlyukevich. Bistable behaviour of a jump-diffusion driven by a periodic stable-like additive process. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3175-3190. doi: 10.3934/dcdsb.2016092
[20]

Wuyuan Jiang. The maximum surplus before ruin in a jump-diffusion insurance risk process with dependence. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3037-3050. doi: 10.3934/dcdsb.2018298

 Impact Factor: 

Tools

Metrics

  • PDF downloads (53)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences