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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

 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.
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 - A, 2015, 35 (9) : 4019-4039. doi: 10.3934/dcds.2015.35.4019
##### References:
 [1] R. Baier, L. Grüne and S. Hafstein, Linear programming based Lyapunov function computation for differential inclusions,, Discrete and Continuous Dynamical Systems Series B, 17 (2012), 33. doi: 10.3934/dcdsb.2012.17.33. Google Scholar [2] H. Ban and W. Kalies, A computational approach to Conley's decomposition theorem,, Journal of Computational and Nonlinear Dynamics, 1 (2006), 312. doi: 10.1115/1.2338651. Google Scholar [3] J. Barnat, J. Chaloupka and J. van de Pol, Distributed algorithms for SCC decomposition,, Journal of Logic and Computation, 21 (2011), 23. doi: 10.1093/logcom/exp003. Google Scholar [4] J. Björnsson, P. Giesl and S. Hafstein, Algorithmic verification of approximations to complete Lyapunov functions,, in Proceedings of the 21st International Symposium on Mathematical Theory of Networks and Systems, (2014), 1181. Google Scholar [5] J. Björnsson, P. Giesl, S. Hafstein, C. M. Kellett and H. Li, Computation of continuous and piecewise affine Lyapunov functions by numerical approximations of the Massera construction,, in Proceedings of the 53rd IEEE Conference on Decision and Control, (2014), 5506. doi: 10.1109/CDC.2014.7040250. Google Scholar [6] C. Conley, Isolated Invariant Sets and the Morse Index,, CBMS Regional Conference Series no. 38, (1978). Google Scholar [7] M. Dellnitz, G. Froyland and O. Junge, The algorithms behind GAIO-set oriented numerical methods for dynamical systems,, in Ergodic theory, (2001), 145. Google Scholar [8] P. Giesl, Construction of Global Lyapunov Functions Using Radial Basis Functions,, no. 1904 in Lecture Notes in Mathematics, (1904). Google Scholar [9] P. Giesl and S. Hafstein, Construction of Lyapunov functions for nonlinear planar systems by linear programming,, Journal of Mathematical Analysis and Applications, 388 (2012), 463. doi: 10.1016/j.jmaa.2011.10.047. Google Scholar [10] P. Giesl and S. Hafstein, Revised CPA method to compute Lyapunov functions for nonlinear systems,, Journal of Mathematical Analysis and Applications, 410 (2014), 292. doi: 10.1016/j.jmaa.2013.08.014. Google Scholar [11] S. Hafstein, An Algorithm for Constructing Lyapunov Functions,, Electronic Journal of Differential Equations Mongraphs, (2007). Google Scholar [12] S. Hafstein, C. M. Kellett and H. Li, Continuous and piecewise affine Lyapunov functions using the Yoshizawa construction,, in Proceedings of the 2014 American Control Conference, (2014), 548. doi: 10.1109/ACC.2014.6858660. Google Scholar [13] M. Hurley, Lyapunov functions and attractors in arbitrary metric spaces,, Proc. Amer. Math. Soc., 126 (1998), 245. doi: 10.1090/S0002-9939-98-04500-6. Google Scholar [14] O. Junge, Mengenorientierte Methoden zur Numerischen Analyse Dynamischer Systeme,, PhD thesis at the University of Paderborn, (2000). Google Scholar [15] W. Kalies, K. Mischaikow and R. VanderVorst, An algorithmic approach to chain recurrence,, Foundations of Computational Mathematics, 5 (2005), 409. doi: 10.1007/s10208-004-0163-9. Google Scholar [16] S. Marinosson, Lyapunov function construction for ordinary differential equations with linear programming,, Dynamical Systems, 17 (2002), 137. doi: 10.1080/0268111011011847. Google Scholar [17] S. Marinosson, Stability Analysis of Nonlinear Systems with Linear Programming: A Lyapunov Functions Based Approach,, PhD thesis, (2002). Google Scholar [18] J. L. Massera, On Liapounoff's conditions of stability,, Annals of Mathematics, 50 (1949), 705. doi: 10.2307/1969558. Google Scholar [19] D. Norton, The fundamental theorem of dynamical systems,, Comment. Math. Univ. Carolinae, 36 (1995), 585. Google Scholar [20] A. Papachristodoulou and S. Prajna, The construction of Lyapunov functions using the sum of squares decomposition,, in Proceedings of the 41st IEEE Conference on Decision and Control, 3 (2002), 3482. doi: 10.1109/CDC.2002.1184414. Google Scholar [21] M. Patrao, Existence of complete Lyapunov functions for semiflows on separable metric spaces,, Far East Journal of Dynamical Systems, 17 (2011), 49. Google Scholar [22] M. Peet and A. Papachristodoulou, A converse sum-of-squares Lyapunov result: An existence proof based on the Picard iteration,, in Proceedings of the 49th IEEE Conference on Decision and Control, (2010), 5949. doi: 10.1109/CDC.2010.5717536. Google Scholar [23] S. Ratschan and Z. She, Providing a basin of attraction to a target region of polynomial systems by computation of Lyapunov-like functions,, SIAM J. Control and Optimization, 48 (2010), 4377. doi: 10.1137/090749955. Google Scholar [24] R. Tarjan, Depth-first search and linear graph algorithms,, SIAM J. Comput., 1 (1972), 146. doi: 10.1137/0201010. Google Scholar [25] A. R. Teel and L. Praly, A smooth Lyapunov function from a class-$\mathcal{KL}$ estimate involving two positive semidefinite functions,, ESAIM Control Optim. Calc. Var., 5 (2000), 313. doi: 10.1051/cocv:2000113. Google Scholar [26] W. Tucker, A rigorous ODE solver and Smale's 14th problem,, Found. Comput. Math., 2 (2002), 53. doi: 10.1007/s002080010018. Google Scholar [27] T. Yoshizawa, On the stability of solutions of a system of differential equations,, Memoirs of the College of Science, 29 (1955), 27. Google Scholar

show all references

##### References:
 [1] R. Baier, L. Grüne and S. Hafstein, Linear programming based Lyapunov function computation for differential inclusions,, Discrete and Continuous Dynamical Systems Series B, 17 (2012), 33. doi: 10.3934/dcdsb.2012.17.33. Google Scholar [2] H. Ban and W. Kalies, A computational approach to Conley's decomposition theorem,, Journal of Computational and Nonlinear Dynamics, 1 (2006), 312. doi: 10.1115/1.2338651. Google Scholar [3] J. Barnat, J. Chaloupka and J. van de Pol, Distributed algorithms for SCC decomposition,, Journal of Logic and Computation, 21 (2011), 23. doi: 10.1093/logcom/exp003. Google Scholar [4] J. Björnsson, P. Giesl and S. Hafstein, Algorithmic verification of approximations to complete Lyapunov functions,, in Proceedings of the 21st International Symposium on Mathematical Theory of Networks and Systems, (2014), 1181. Google Scholar [5] J. Björnsson, P. Giesl, S. Hafstein, C. M. Kellett and H. Li, Computation of continuous and piecewise affine Lyapunov functions by numerical approximations of the Massera construction,, in Proceedings of the 53rd IEEE Conference on Decision and Control, (2014), 5506. doi: 10.1109/CDC.2014.7040250. Google Scholar [6] C. Conley, Isolated Invariant Sets and the Morse Index,, CBMS Regional Conference Series no. 38, (1978). Google Scholar [7] M. Dellnitz, G. Froyland and O. Junge, The algorithms behind GAIO-set oriented numerical methods for dynamical systems,, in Ergodic theory, (2001), 145. Google Scholar [8] P. Giesl, Construction of Global Lyapunov Functions Using Radial Basis Functions,, no. 1904 in Lecture Notes in Mathematics, (1904). Google Scholar [9] P. Giesl and S. Hafstein, Construction of Lyapunov functions for nonlinear planar systems by linear programming,, Journal of Mathematical Analysis and Applications, 388 (2012), 463. doi: 10.1016/j.jmaa.2011.10.047. Google Scholar [10] P. Giesl and S. Hafstein, Revised CPA method to compute Lyapunov functions for nonlinear systems,, Journal of Mathematical Analysis and Applications, 410 (2014), 292. doi: 10.1016/j.jmaa.2013.08.014. Google Scholar [11] S. Hafstein, An Algorithm for Constructing Lyapunov Functions,, Electronic Journal of Differential Equations Mongraphs, (2007). Google Scholar [12] S. Hafstein, C. M. Kellett and H. Li, Continuous and piecewise affine Lyapunov functions using the Yoshizawa construction,, in Proceedings of the 2014 American Control Conference, (2014), 548. doi: 10.1109/ACC.2014.6858660. Google Scholar [13] M. Hurley, Lyapunov functions and attractors in arbitrary metric spaces,, Proc. Amer. Math. Soc., 126 (1998), 245. doi: 10.1090/S0002-9939-98-04500-6. Google Scholar [14] O. Junge, Mengenorientierte Methoden zur Numerischen Analyse Dynamischer Systeme,, PhD thesis at the University of Paderborn, (2000). Google Scholar [15] W. Kalies, K. Mischaikow and R. VanderVorst, An algorithmic approach to chain recurrence,, Foundations of Computational Mathematics, 5 (2005), 409. doi: 10.1007/s10208-004-0163-9. Google Scholar [16] S. Marinosson, Lyapunov function construction for ordinary differential equations with linear programming,, Dynamical Systems, 17 (2002), 137. doi: 10.1080/0268111011011847. Google Scholar [17] S. Marinosson, Stability Analysis of Nonlinear Systems with Linear Programming: A Lyapunov Functions Based Approach,, PhD thesis, (2002). Google Scholar [18] J. L. Massera, On Liapounoff's conditions of stability,, Annals of Mathematics, 50 (1949), 705. doi: 10.2307/1969558. Google Scholar [19] D. Norton, The fundamental theorem of dynamical systems,, Comment. Math. Univ. Carolinae, 36 (1995), 585. Google Scholar [20] A. Papachristodoulou and S. Prajna, The construction of Lyapunov functions using the sum of squares decomposition,, in Proceedings of the 41st IEEE Conference on Decision and Control, 3 (2002), 3482. doi: 10.1109/CDC.2002.1184414. Google Scholar [21] M. Patrao, Existence of complete Lyapunov functions for semiflows on separable metric spaces,, Far East Journal of Dynamical Systems, 17 (2011), 49. Google Scholar [22] M. Peet and A. Papachristodoulou, A converse sum-of-squares Lyapunov result: An existence proof based on the Picard iteration,, in Proceedings of the 49th IEEE Conference on Decision and Control, (2010), 5949. doi: 10.1109/CDC.2010.5717536. Google Scholar [23] S. Ratschan and Z. She, Providing a basin of attraction to a target region of polynomial systems by computation of Lyapunov-like functions,, SIAM J. Control and Optimization, 48 (2010), 4377. doi: 10.1137/090749955. Google Scholar [24] R. Tarjan, Depth-first search and linear graph algorithms,, SIAM J. Comput., 1 (1972), 146. doi: 10.1137/0201010. Google Scholar [25] A. R. Teel and L. Praly, A smooth Lyapunov function from a class-$\mathcal{KL}$ estimate involving two positive semidefinite functions,, ESAIM Control Optim. Calc. Var., 5 (2000), 313. doi: 10.1051/cocv:2000113. Google Scholar [26] W. Tucker, A rigorous ODE solver and Smale's 14th problem,, Found. Comput. Math., 2 (2002), 53. doi: 10.1007/s002080010018. Google Scholar [27] T. Yoshizawa, On the stability of solutions of a system of differential equations,, Memoirs of the College of Science, 29 (1955), 27. Google Scholar
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