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September  2015, 35(9): 4019-4039. doi: 10.3934/dcds.2015.35.4019

Computation of Lyapunov functions for systems with multiple local attractors

Jóhann Björnsson 1, , Peter Giesl 2, , Sigurdur F. Hafstein 1, and  Christopher M. Kellett 3,

1. 

School of Science and Engineering, Reykjavik University, Menntavegi 1, Reykjavik, IS-101, Iceland, Iceland
2. 

Department of Mathematics, University of Sussex, Falmer BN1 9QH
3. 

School of Electrical Engineering and Computer Science, University of Newcastle, Callaghan, New South Wales 2308, Australia

Received  June 2014 Revised  October 2014 Published  April 2015

We present a novel method to compute Lyapunov functions for continuous-time systems with multiple local attractors. In the proposed method one first computes an outer approximation of the local attractors using a graph-theoretic approach. Then a candidate Lyapunov function is computed using a Massera-like construction adapted to multiple local attractors. In the final step this candidate Lyapunov function is interpolated over the simplices of a simplicial complex and, by checking certain inequalities at the vertices of the complex, we can identify the region in which the Lyapunov function is decreasing along system trajectories. The resulting Lyapunov function gives information on the qualitative behavior of the dynamics, including lower bounds on the basins of attraction of the individual local attractors. We develop the theory in detail and present numerical examples demonstrating the applicability of our method.
Keywords: numerical method., asymptotic stability, multiple local attractors, Lyapunov function, dynamical system.
Mathematics Subject Classification: Primary: 37B25, 37M99; Secondary: 37C1.
Citation: Jóhann Björnsson, Peter Giesl, Sigurdur F. Hafstein, Christopher M. Kellett. Computation of Lyapunov functions for systems with multiple local attractors. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4019-4039. doi: 10.3934/dcds.2015.35.4019
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show all references

References:
[1]

Discrete and Continuous Dynamical Systems Series B, 17 (2012), 33-56. doi: 10.3934/dcdsb.2012.17.33.  Google Scholar

[2]

Journal of Computational and Nonlinear Dynamics, 1 (2006), 312-319. doi: 10.1115/1.2338651.  Google Scholar

[3]

Journal of Logic and Computation, 21 (2011), 23-44. doi: 10.1093/logcom/exp003.  Google Scholar

[4]

in Proceedings of the 21st International Symposium on Mathematical Theory of Networks and Systems, Groningen, The Netherlands, (2014), 1181-1188 (no. 0180). Google Scholar

[5]

in Proceedings of the 53rd IEEE Conference on Decision and Control, Los Angeles, California, USA, (2014), 5506-5511. doi: 10.1109/CDC.2014.7040250.  Google Scholar

[6]

CBMS Regional Conference Series no. 38, American Mathematical Society, 1978.  Google Scholar

[7]

in Ergodic theory, analysis, and efficient simulation of dynamical systems, Springer, Berlin, (2001), 145-174, 805-807.  Google Scholar

[8]

no. 1904 in Lecture Notes in Mathematics, Springer, 2007.  Google Scholar

[9]

Journal of Mathematical Analysis and Applications, 388 (2012), 463-479. doi: 10.1016/j.jmaa.2011.10.047.  Google Scholar

[10]

Journal of Mathematical Analysis and Applications, 410 (2014), 292-306. doi: 10.1016/j.jmaa.2013.08.014.  Google Scholar

[11]

Electronic Journal of Differential Equations Mongraphs, 2007. Google Scholar

[12]

in Proceedings of the 2014 American Control Conference, Portland, Oregon, USA, (2014), 548-553 (no. 0170). doi: 10.1109/ACC.2014.6858660.  Google Scholar

[13]

Proc. Amer. Math. Soc., 126 (1998), 245-256. doi: 10.1090/S0002-9939-98-04500-6.  Google Scholar

[14]

PhD thesis at the University of Paderborn, Germany, Shaker, 2000. Google Scholar

[15]

Foundations of Computational Mathematics, 5 (2005), 409-449. doi: 10.1007/s10208-004-0163-9.  Google Scholar

[16]

Dynamical Systems, 17 (2002), 137-150. doi: 10.1080/0268111011011847.  Google Scholar

[17]

PhD thesis, Gerhard-Mercator-University, Duisburg, Germany, 2002. Google Scholar

[18]

Annals of Mathematics, 50 (1949), 705-721. doi: 10.2307/1969558.  Google Scholar

[19]

Comment. Math. Univ. Carolinae, 36 (1995), 585-597.  Google Scholar

[20]

in Proceedings of the 41st IEEE Conference on Decision and Control, 3, Las Vegas, Nevada, USA, (2002), 3482-3487. doi: 10.1109/CDC.2002.1184414.  Google Scholar

[21]

Far East Journal of Dynamical Systems, 17 (2011), 49-54.  Google Scholar

[22]

in Proceedings of the 49th IEEE Conference on Decision and Control, Atlanta, Georgia, USA, (2010), 5949-5954. doi: 10.1109/CDC.2010.5717536.  Google Scholar

[23]

SIAM J. Control and Optimization, 48 (2010), 4377-4394. doi: 10.1137/090749955.  Google Scholar

[24]

SIAM J. Comput., 1 (1972), 146-160. doi: 10.1137/0201010.  Google Scholar

[25]

ESAIM Control Optim. Calc. Var., 5 (2000), 313-367. doi: 10.1051/cocv:2000113.  Google Scholar

[26]

Found. Comput. Math., 2 (2002), 53-117. doi: 10.1007/s002080010018.  Google Scholar

[27]

Memoirs of the College of Science, University of Kyoto, Series A: Mathematics, 29 (1955), 27-33.  Google Scholar

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