# American Institue of Mathematical Sciences

2015, 2(2): 227-246. doi: 10.3934/jcd.2015004

## Computing continuous and piecewise affine lyapunov functions for nonlinear systems

 1 School of Science and Engineering, Reykjavik University, Menntavegi 1, Reykjavik, IS-101 2 School of Electrical Engineering and Computer Science, University of Newcastle, Callaghan, New South Wales 2308 3 School of Mathematics and Physics, Chinese University of Geosciences (Wuhan), 430074, Wuhan

Received  June 2015 Revised  March 2016 Published  May 2016

We present a numerical technique for the computation of a Lyapunov function for nonlinear systems with an asymptotically stable equilibrium point. The proposed approach constructs a partition of the state space, called a triangulation, and then computes values at the vertices of the triangulation using a Lyapunov function from a classical converse Lyapunov theorem due to Yoshizawa. A simple interpolation of the vertex values then yields a Continuous and Piecewise Affine (CPA) function. Verification that the obtained CPA function is a Lyapunov function is shown to be equivalent to verification of several simple inequalities. Numerical examples are presented demonstrating different aspects of the proposed method.
Citation: Sigurdur F. Hafstein, Christopher M. Kellett, Huijuan Li. Computing continuous and piecewise affine lyapunov functions for nonlinear systems. Journal of Computational Dynamics, 2015, 2 (2) : 227-246. doi: 10.3934/jcd.2015004
##### 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. [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. [3] 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. [4] 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. [5] P. Giesl, Construction of Global Lyapunov Functions Using Radial Basis Functions,, Number 1904 in Lecture Notes in Mathematics. Springer, (1904). [6] P. Giesl and S. Hafstein, Existence of piecewise affine Lyapunov functions in two dimensions,, J. Math. Anal. Appl., 371 (2010), 233. doi: 10.1016/j.jmaa.2010.05.009. [7] 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. [8] P. Giesl and S. Hafstein, Existence of piecewise affine Lyapunov functions in arbitrary dimensions,, Discrete and Contin. Dyn. Syst., 32 (2012), 3539. doi: 10.3934/dcds.2012.32.3539. [9] 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. [10] P. Giesl and S. Hafstein, Computation and verification of Lyapunov functions,, SIAM J. Appl. Dyn. Syst., 14 (2015), 1663. doi: 10.1137/140988802. [11] P. Giesl and S. Hafstein, Review on computational methods for Lyapunov functions,, Discrete and Continuous Dynamical Systems Series B, 20 (2015), 2291. doi: 10.3934/dcdsb.2015.20.2291. [12] S. Hafstein, An Algorithm for Constructing Lyapunov Functions,, Electronic Journal of Differential Equations Mongraphs, (2007). [13] S. Hafstein, C. M. Kellett and H. Li, Continuous and piecewise affine Lyapunov functions using the Yoshizawa construction,, In Proceedings of the American Control Conference, (2014), 548. doi: 10.1109/ACC.2014.6858660. [14] W. Hahn, Stability of Motion,, Springer-Verlag, (1967). [15] T. Johansen, Computation of Lyapunov functions for smooth nonlinear systems using convex optimization,, Automatica, 36 (2000), 1617. doi: 10.1016/S0005-1098(00)00088-1. [16] 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. [17] C. M. Kellett, A compendium of comparsion function results,, Mathematics of Controls, 26 (2014), 339. doi: 10.1007/s00498-014-0128-8. [18] C. M. Kellett, Classical converse theorems in Lyapunov's second method,, Discrete and Continuous Dynamical Systems Series B, 20 (2015), 2333. doi: 10.3934/dcdsb.2015.20.2333. [19] J. Kurzweil, On the inversion of Ljapunov's second theorem on stability of motion,, Chechoslovak Mathematics Journal, 81 (1956), 217. [20] H. Li, S. Hafstein and C. M. Kellett, Computation of continuous and piecewise affine Lyapunov functions for discrete-time systems,, J Differ Equ Appl, 21 (2015), 486. doi: 10.1080/10236198.2015.1025069. [21] A. M. Lyapunov, The general problem of the stability of motion,, Math. Soc. of Kharkov, 55 (1992), 521. doi: 10.1080/00207179208934253. [22] S. Marinosson, Lyapunov function construction for ordinary differential equations with linear programming,, Dynamical Systems, 17 (2002), 137. doi: 10.1080/0268111011011847. [23] J. L. Massera, On Liapounoff's conditions of stability,, Annals of Mathematics, 50 (1949), 705. doi: 10.2307/1969558. [24] 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. [25] M. Peet and A. Papachristodoulou, A converse sum of squares Lyapunov result with a degree bound,, IEEE Transactions on Automatic Control, 57 (2012), 2281. doi: 10.1109/TAC.2012.2190163. [26] N. Rouche, P. Habets and M. Laloy, Stability Theory by Liapunov's Direct Method,, Springer-Verlag, (1977). [27] E. D. Sontag, Comments on integral variants of ISS,, Systems and Control Letters, 34 (1998), 93. doi: 10.1016/S0167-6911(98)00003-6. [28] 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. [29] T. Yoshizawa, On the stability of solutions of a system of differential equations,, Memoirs of the College of Science, 29 (1955), 27. [30] T. Yoshizawa, Stability Theory by Liapunov's Second Method,, Mathematical Society of Japan, (1966).

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##### 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. [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. [3] 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. [4] 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. [5] P. Giesl, Construction of Global Lyapunov Functions Using Radial Basis Functions,, Number 1904 in Lecture Notes in Mathematics. Springer, (1904). [6] P. Giesl and S. Hafstein, Existence of piecewise affine Lyapunov functions in two dimensions,, J. Math. Anal. Appl., 371 (2010), 233. doi: 10.1016/j.jmaa.2010.05.009. [7] 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. [8] P. Giesl and S. Hafstein, Existence of piecewise affine Lyapunov functions in arbitrary dimensions,, Discrete and Contin. Dyn. Syst., 32 (2012), 3539. doi: 10.3934/dcds.2012.32.3539. [9] 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. [10] P. Giesl and S. Hafstein, Computation and verification of Lyapunov functions,, SIAM J. Appl. Dyn. Syst., 14 (2015), 1663. doi: 10.1137/140988802. [11] P. Giesl and S. Hafstein, Review on computational methods for Lyapunov functions,, Discrete and Continuous Dynamical Systems Series B, 20 (2015), 2291. doi: 10.3934/dcdsb.2015.20.2291. [12] S. Hafstein, An Algorithm for Constructing Lyapunov Functions,, Electronic Journal of Differential Equations Mongraphs, (2007). [13] S. Hafstein, C. M. Kellett and H. Li, Continuous and piecewise affine Lyapunov functions using the Yoshizawa construction,, In Proceedings of the American Control Conference, (2014), 548. doi: 10.1109/ACC.2014.6858660. [14] W. Hahn, Stability of Motion,, Springer-Verlag, (1967). [15] T. Johansen, Computation of Lyapunov functions for smooth nonlinear systems using convex optimization,, Automatica, 36 (2000), 1617. doi: 10.1016/S0005-1098(00)00088-1. [16] 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. [17] C. M. Kellett, A compendium of comparsion function results,, Mathematics of Controls, 26 (2014), 339. doi: 10.1007/s00498-014-0128-8. [18] C. M. Kellett, Classical converse theorems in Lyapunov's second method,, Discrete and Continuous Dynamical Systems Series B, 20 (2015), 2333. doi: 10.3934/dcdsb.2015.20.2333. [19] J. Kurzweil, On the inversion of Ljapunov's second theorem on stability of motion,, Chechoslovak Mathematics Journal, 81 (1956), 217. [20] H. Li, S. Hafstein and C. M. Kellett, Computation of continuous and piecewise affine Lyapunov functions for discrete-time systems,, J Differ Equ Appl, 21 (2015), 486. doi: 10.1080/10236198.2015.1025069. [21] A. M. Lyapunov, The general problem of the stability of motion,, Math. Soc. of Kharkov, 55 (1992), 521. doi: 10.1080/00207179208934253. [22] S. Marinosson, Lyapunov function construction for ordinary differential equations with linear programming,, Dynamical Systems, 17 (2002), 137. doi: 10.1080/0268111011011847. [23] J. L. Massera, On Liapounoff's conditions of stability,, Annals of Mathematics, 50 (1949), 705. doi: 10.2307/1969558. [24] 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. [25] M. Peet and A. Papachristodoulou, A converse sum of squares Lyapunov result with a degree bound,, IEEE Transactions on Automatic Control, 57 (2012), 2281. doi: 10.1109/TAC.2012.2190163. [26] N. Rouche, P. Habets and M. Laloy, Stability Theory by Liapunov's Direct Method,, Springer-Verlag, (1977). [27] E. D. Sontag, Comments on integral variants of ISS,, Systems and Control Letters, 34 (1998), 93. doi: 10.1016/S0167-6911(98)00003-6. [28] 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. [29] T. Yoshizawa, On the stability of solutions of a system of differential equations,, Memoirs of the College of Science, 29 (1955), 27. [30] T. Yoshizawa, Stability Theory by Liapunov's Second Method,, Mathematical Society of Japan, (1966).
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