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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 and Continuous Dynamical Systems, 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), 33-56. doi: 10.3934/dcdsb.2012.17.33.

[2]

H. Ban and W. Kalies, A computational approach to Conley's decomposition theorem, Journal of Computational and Nonlinear Dynamics, 1 (2006), 312-319. doi: 10.1115/1.2338651.

[3]

J. Barnat, J. Chaloupka and J. van de Pol, Distributed algorithms for SCC decomposition, Journal of Logic and Computation, 21 (2011), 23-44. doi: 10.1093/logcom/exp003.

[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, Groningen, The Netherlands, (2014), 1181-1188 (no. 0180).

[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, Los Angeles, California, USA, (2014), 5506-5511. doi: 10.1109/CDC.2014.7040250.

[6]

C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series no. 38, American Mathematical Society, 1978.

[7]

M. Dellnitz, G. Froyland and O. Junge, The algorithms behind GAIO-set oriented numerical methods for dynamical systems, in Ergodic theory, analysis, and efficient simulation of dynamical systems, Springer, Berlin, (2001), 145-174, 805-807.

[8]

P. Giesl, Construction of Global Lyapunov Functions Using Radial Basis Functions, no. 1904 in Lecture Notes in Mathematics, Springer, 2007.

[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), 463-479. doi: 10.1016/j.jmaa.2011.10.047.

[10]

P. Giesl and S. Hafstein, Revised CPA method to compute Lyapunov functions for nonlinear systems, Journal of Mathematical Analysis and Applications, 410 (2014), 292-306. doi: 10.1016/j.jmaa.2013.08.014.

[11]

S. Hafstein, An Algorithm for Constructing Lyapunov Functions, Electronic Journal of Differential Equations Mongraphs, 2007.

[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, Portland, Oregon, USA, (2014), 548-553 (no. 0170). doi: 10.1109/ACC.2014.6858660.

[13]

M. Hurley, Lyapunov functions and attractors in arbitrary metric spaces, Proc. Amer. Math. Soc., 126 (1998), 245-256. doi: 10.1090/S0002-9939-98-04500-6.

[14]

O. Junge, Mengenorientierte Methoden zur Numerischen Analyse Dynamischer Systeme, PhD thesis at the University of Paderborn, Germany, Shaker, 2000.

[15]

W. Kalies, K. Mischaikow and R. VanderVorst, An algorithmic approach to chain recurrence, Foundations of Computational Mathematics, 5 (2005), 409-449. doi: 10.1007/s10208-004-0163-9.

[16]

S. Marinosson, Lyapunov function construction for ordinary differential equations with linear programming, Dynamical Systems, 17 (2002), 137-150. doi: 10.1080/0268111011011847.

[17]

S. Marinosson, Stability Analysis of Nonlinear Systems with Linear Programming: A Lyapunov Functions Based Approach, PhD thesis, Gerhard-Mercator-University, Duisburg, Germany, 2002.

[18]

J. L. Massera, On Liapounoff's conditions of stability, Annals of Mathematics, 50 (1949), 705-721. doi: 10.2307/1969558.

[19]

D. Norton, The fundamental theorem of dynamical systems, Comment. Math. Univ. Carolinae, 36 (1995), 585-597.

[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, Las Vegas, Nevada, USA, (2002), 3482-3487. doi: 10.1109/CDC.2002.1184414.

[21]

M. Patrao, Existence of complete Lyapunov functions for semiflows on separable metric spaces, Far East Journal of Dynamical Systems, 17 (2011), 49-54.

[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, Atlanta, Georgia, USA, (2010), 5949-5954. doi: 10.1109/CDC.2010.5717536.

[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), 4377-4394. doi: 10.1137/090749955.

[24]

R. Tarjan, Depth-first search and linear graph algorithms, SIAM J. Comput., 1 (1972), 146-160. doi: 10.1137/0201010.

[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), 313-367. doi: 10.1051/cocv:2000113.

[26]

W. Tucker, A rigorous ODE solver and Smale's 14th problem, Found. Comput. Math., 2 (2002), 53-117. doi: 10.1007/s002080010018.

[27]

T. Yoshizawa, On the stability of solutions of a system of differential equations, Memoirs of the College of Science, University of Kyoto, Series A: Mathematics, 29 (1955), 27-33.

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), 33-56. doi: 10.3934/dcdsb.2012.17.33.

[2]

H. Ban and W. Kalies, A computational approach to Conley's decomposition theorem, Journal of Computational and Nonlinear Dynamics, 1 (2006), 312-319. doi: 10.1115/1.2338651.

[3]

J. Barnat, J. Chaloupka and J. van de Pol, Distributed algorithms for SCC decomposition, Journal of Logic and Computation, 21 (2011), 23-44. doi: 10.1093/logcom/exp003.

[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, Groningen, The Netherlands, (2014), 1181-1188 (no. 0180).

[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, Los Angeles, California, USA, (2014), 5506-5511. doi: 10.1109/CDC.2014.7040250.

[6]

C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series no. 38, American Mathematical Society, 1978.

[7]

M. Dellnitz, G. Froyland and O. Junge, The algorithms behind GAIO-set oriented numerical methods for dynamical systems, in Ergodic theory, analysis, and efficient simulation of dynamical systems, Springer, Berlin, (2001), 145-174, 805-807.

[8]

P. Giesl, Construction of Global Lyapunov Functions Using Radial Basis Functions, no. 1904 in Lecture Notes in Mathematics, Springer, 2007.

[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), 463-479. doi: 10.1016/j.jmaa.2011.10.047.

[10]

P. Giesl and S. Hafstein, Revised CPA method to compute Lyapunov functions for nonlinear systems, Journal of Mathematical Analysis and Applications, 410 (2014), 292-306. doi: 10.1016/j.jmaa.2013.08.014.

[11]

S. Hafstein, An Algorithm for Constructing Lyapunov Functions, Electronic Journal of Differential Equations Mongraphs, 2007.

[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, Portland, Oregon, USA, (2014), 548-553 (no. 0170). doi: 10.1109/ACC.2014.6858660.

[13]

M. Hurley, Lyapunov functions and attractors in arbitrary metric spaces, Proc. Amer. Math. Soc., 126 (1998), 245-256. doi: 10.1090/S0002-9939-98-04500-6.

[14]

O. Junge, Mengenorientierte Methoden zur Numerischen Analyse Dynamischer Systeme, PhD thesis at the University of Paderborn, Germany, Shaker, 2000.

[15]

W. Kalies, K. Mischaikow and R. VanderVorst, An algorithmic approach to chain recurrence, Foundations of Computational Mathematics, 5 (2005), 409-449. doi: 10.1007/s10208-004-0163-9.

[16]

S. Marinosson, Lyapunov function construction for ordinary differential equations with linear programming, Dynamical Systems, 17 (2002), 137-150. doi: 10.1080/0268111011011847.

[17]

S. Marinosson, Stability Analysis of Nonlinear Systems with Linear Programming: A Lyapunov Functions Based Approach, PhD thesis, Gerhard-Mercator-University, Duisburg, Germany, 2002.

[18]

J. L. Massera, On Liapounoff's conditions of stability, Annals of Mathematics, 50 (1949), 705-721. doi: 10.2307/1969558.

[19]

D. Norton, The fundamental theorem of dynamical systems, Comment. Math. Univ. Carolinae, 36 (1995), 585-597.

[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, Las Vegas, Nevada, USA, (2002), 3482-3487. doi: 10.1109/CDC.2002.1184414.

[21]

M. Patrao, Existence of complete Lyapunov functions for semiflows on separable metric spaces, Far East Journal of Dynamical Systems, 17 (2011), 49-54.

[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, Atlanta, Georgia, USA, (2010), 5949-5954. doi: 10.1109/CDC.2010.5717536.

[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), 4377-4394. doi: 10.1137/090749955.

[24]

R. Tarjan, Depth-first search and linear graph algorithms, SIAM J. Comput., 1 (1972), 146-160. doi: 10.1137/0201010.

[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), 313-367. doi: 10.1051/cocv:2000113.

[26]

W. Tucker, A rigorous ODE solver and Smale's 14th problem, Found. Comput. Math., 2 (2002), 53-117. doi: 10.1007/s002080010018.

[27]

T. Yoshizawa, On the stability of solutions of a system of differential equations, Memoirs of the College of Science, University of Kyoto, Series A: Mathematics, 29 (1955), 27-33.

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