# American Institute of Mathematical Sciences

October  2015, 20(8): 2453-2476. doi: 10.3934/dcdsb.2015.20.2453

## Grid refinement in the construction of Lyapunov functions using radial basis functions

 1 Department of Mathematics, University of Sussex, Falmer BN1 9QH, United Kingdom

Received  September 2014 Revised  January 2015 Published  August 2015

Lyapunov functions are a main tool to determine the domain of attraction of equilibria in dynamical systems. Recently, several methods have been presented to construct a Lyapunov function for a given system. In this paper, we improve the construction method for Lyapunov functions using Radial Basis Functions. We combine this method with a new grid refinement algorithm based on Voronoi diagrams. Starting with a coarse grid and applying the refinement algorithm, we thus manage to reduce the number of data points needed to construct Lyapunov functions. Finally, we give numerical examples to illustrate our algorithms.
Citation: Najla Mohammed, Peter Giesl. Grid refinement in the construction of Lyapunov functions using radial basis functions. Discrete and Continuous Dynamical Systems - B, 2015, 20 (8) : 2453-2476. doi: 10.3934/dcdsb.2015.20.2453
##### References:
 [1] M. Berg, O. Cheong, M. Kerveld and M. Overmars, Computational Geometry: Algorithms and Applications, Springer-Verlag, Berlin, 2008. [2] M. D. Buhmann, Radial basis functions, in Acta Numerica, 2000, Acta Numer., 9, Cambridge Univ. Press, Cambridge, 2000, 1-38. doi: 10.1017/S0962492900000015. [3] F. Camilli, L. Grüne and F. Wirth, A generalization of Zubov's method to perturbed systems, SIAM J. Control Optim., 40 (2001), 496-515. doi: 10.1137/S036301299936316X. [4] M. Dellnitz and O. Junge, Set oriented numerical methods for dynamical systems, in Handbook of Dynamical Systems, Vol. 2, North-Holland, Amsterdam, 2002, 221-264. doi: 10.1016/S1874-575X(02)80026-1. [5] M. Floater and A. Iske, Multistep scattered data interpolation using compactly supported Radial Basis Functions, J. Comput. Appl. Math., 73 (1996), 65-78. doi: 10.1016/0377-0427(96)00035-0. [6] P. Giesl, Construction of Global Lyapunov Functions Using Radial Basis Functions, Lecture Notes in Math., 1904, Springer, 2007. [7] P. Giesl, Construction of a local and global Lyapunov function using Radial Basis Functions, IMA J. Appl. Math., 73 (2008), 782-802. doi: 10.1093/imamat/hxn018. [8] P. Giesl and S. Hafstein, Revised CPA method to compute Lyapunov functions for nonlinear systems, J. Math. Anal. Appl., 410 (2014), 292-306. doi: 10.1016/j.jmaa.2013.08.014. [9] P. Giesl and S. Hafstein, Review on computational methods for Lyapunov functions, Discrete and Continuous Dynamical Systems - Series B, 8 (2015), 2291-2331. doi: 10.3934/dcdsb.2015.20.2291. [10] P. Giesl and H. Wendland, Meshless collocation: Error estimates with application to Dynamical Systems, SIAM J. Numer. Anal., 45 (2007), 1723-1741. doi: 10.1137/060658813. [11] L. Grüne, An adaptive grid scheme for the discrete Hamilton-Jacobi-Bellman equation, Numer. Math., 75 (1997), 319-337. doi: 10.1007/s002110050241. [12] L. Grüne, Asymptotic Behavior of Dynamical and Control Systems Under Perturbation and Discretization, Lecture Notes in Mathematics, 1783, Springer-Verlag, Berlin, 2002. doi: 10.1007/b83677. [13] S. Hafstein, A constructive converse Lyapunov theorem on exponential stability, Discrete and Continuous Dynamical Systems - Series A, 10 (2004), 657-678. doi: 10.3934/dcds.2004.10.657. [14] S. Hafstein, An algorithm for constructing Lyapunov functions, Monograph. Electron. J. Diff. Eqns., (2007), 101pp. [15] C. S. Hsu, Cell-to-cell Mapping, Applied Mathematical Sciences, 64, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4757-3892-6. [16] A. Iske, On the construction of kernel-based adaptive particle methods in numerical flow simulation, in Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws, Notes Numer. Fluid Mech. Multidiscip. Des., 120, Springer, Heidelberg, 2013, 197-221. doi: 10.1007/978-3-642-33221-0_12. [17] S. Iyengar, K. Boroojeni and N. Balakrishnan, Mathematical Theories of Distributed Sensor Networks, Springer, New York, 2014. doi: 10.1007/978-1-4419-8420-3. [18] Z. Jian, Development of Strong Form Methods with Applications in Computational Mechanics, PhD thesis, National University of Singapore, Singapore, 2008. [19] C. Kellett, Classical converse theorems in Lyapunov’s second method, Discrete and Continuous Dynamical Systems - Series B, 8 (2015), 2333-2360. doi: 10.3934/dcdsb.2015.20.2333. [20] R. Klein, Concrete and Abstract Voronoi Diagrams, Lecture Notes in Computer Science, Springer-Verlag, Berlin, 1989. doi: 10.1007/3-540-52055-4. [21] J. Massera, On Liapounoff's conditions of stability, Ann. of Math., 50 (1949), 705-721. [22] A. Papachristodoulou, J. Anderson, G. Valmorbida, S. Pranja, P. Seiler and P. Parrilo, SOSTOOLS: Sum of Squares Optimization Toolbox for MATLAB, User's guide. Version 3.00 edition, 2013. [23] P. Parrilo, Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimiza, PhD thesis, California Institute of Technology Pasadena, California, 2000. [24] F. Preparata and M. Shamos, Computational Geometry, Texts and Monographs in Computer Science, Springer-Verlag, New York, 1985. doi: 10.1007/978-1-4612-1098-6. [25] J. Ruppert, A Delaunay refinement algorithm for quality 2-dimensional mesh generation, J. Approx. Theory, 18 (1995), 548-585. doi: 10.1006/jagm.1995.1021. [26] R. Sibson, Development of strong form methods with applications in computational mechanics, in Interpolating Multivariate Data, Chapter 2 (ed. V. Barnett), John Wiley and Sons, New York, 1981. [27] H. Wendland, Error estimates for interpolation by compactly supported Radial Basis Functions of minimal degree, J. Approx. Theory, 93 (1998), 258-272. doi: 10.1006/jath.1997.3137. [28] H. Wendland, Scattered Data Approximation, Cambridge Monographs on Applied and Computational Mathematics, 17, Cambridge University Press, Cambridge, 2005. [29] X. Zhang, R. Ding and Y. Li, Adaptive RPIM meshless method, in Proceedings of the 2011 International Conference on Multimedia Technology (ICMT), IEEE, 2011, 2388-2392.

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##### References:
 [1] M. Berg, O. Cheong, M. Kerveld and M. Overmars, Computational Geometry: Algorithms and Applications, Springer-Verlag, Berlin, 2008. [2] M. D. Buhmann, Radial basis functions, in Acta Numerica, 2000, Acta Numer., 9, Cambridge Univ. Press, Cambridge, 2000, 1-38. doi: 10.1017/S0962492900000015. [3] F. Camilli, L. Grüne and F. Wirth, A generalization of Zubov's method to perturbed systems, SIAM J. Control Optim., 40 (2001), 496-515. doi: 10.1137/S036301299936316X. [4] M. Dellnitz and O. Junge, Set oriented numerical methods for dynamical systems, in Handbook of Dynamical Systems, Vol. 2, North-Holland, Amsterdam, 2002, 221-264. doi: 10.1016/S1874-575X(02)80026-1. [5] M. Floater and A. Iske, Multistep scattered data interpolation using compactly supported Radial Basis Functions, J. Comput. Appl. Math., 73 (1996), 65-78. doi: 10.1016/0377-0427(96)00035-0. [6] P. Giesl, Construction of Global Lyapunov Functions Using Radial Basis Functions, Lecture Notes in Math., 1904, Springer, 2007. [7] P. Giesl, Construction of a local and global Lyapunov function using Radial Basis Functions, IMA J. Appl. Math., 73 (2008), 782-802. doi: 10.1093/imamat/hxn018. [8] P. Giesl and S. Hafstein, Revised CPA method to compute Lyapunov functions for nonlinear systems, J. Math. Anal. Appl., 410 (2014), 292-306. doi: 10.1016/j.jmaa.2013.08.014. [9] P. Giesl and S. Hafstein, Review on computational methods for Lyapunov functions, Discrete and Continuous Dynamical Systems - Series B, 8 (2015), 2291-2331. doi: 10.3934/dcdsb.2015.20.2291. [10] P. Giesl and H. Wendland, Meshless collocation: Error estimates with application to Dynamical Systems, SIAM J. Numer. Anal., 45 (2007), 1723-1741. doi: 10.1137/060658813. [11] L. Grüne, An adaptive grid scheme for the discrete Hamilton-Jacobi-Bellman equation, Numer. Math., 75 (1997), 319-337. doi: 10.1007/s002110050241. [12] L. Grüne, Asymptotic Behavior of Dynamical and Control Systems Under Perturbation and Discretization, Lecture Notes in Mathematics, 1783, Springer-Verlag, Berlin, 2002. doi: 10.1007/b83677. [13] S. Hafstein, A constructive converse Lyapunov theorem on exponential stability, Discrete and Continuous Dynamical Systems - Series A, 10 (2004), 657-678. doi: 10.3934/dcds.2004.10.657. [14] S. Hafstein, An algorithm for constructing Lyapunov functions, Monograph. Electron. J. Diff. Eqns., (2007), 101pp. [15] C. S. Hsu, Cell-to-cell Mapping, Applied Mathematical Sciences, 64, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4757-3892-6. [16] A. Iske, On the construction of kernel-based adaptive particle methods in numerical flow simulation, in Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws, Notes Numer. Fluid Mech. Multidiscip. Des., 120, Springer, Heidelberg, 2013, 197-221. doi: 10.1007/978-3-642-33221-0_12. [17] S. Iyengar, K. Boroojeni and N. Balakrishnan, Mathematical Theories of Distributed Sensor Networks, Springer, New York, 2014. doi: 10.1007/978-1-4419-8420-3. [18] Z. Jian, Development of Strong Form Methods with Applications in Computational Mechanics, PhD thesis, National University of Singapore, Singapore, 2008. [19] C. Kellett, Classical converse theorems in Lyapunov’s second method, Discrete and Continuous Dynamical Systems - Series B, 8 (2015), 2333-2360. doi: 10.3934/dcdsb.2015.20.2333. [20] R. Klein, Concrete and Abstract Voronoi Diagrams, Lecture Notes in Computer Science, Springer-Verlag, Berlin, 1989. doi: 10.1007/3-540-52055-4. [21] J. Massera, On Liapounoff's conditions of stability, Ann. of Math., 50 (1949), 705-721. [22] A. Papachristodoulou, J. Anderson, G. Valmorbida, S. Pranja, P. Seiler and P. Parrilo, SOSTOOLS: Sum of Squares Optimization Toolbox for MATLAB, User's guide. Version 3.00 edition, 2013. [23] P. Parrilo, Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimiza, PhD thesis, California Institute of Technology Pasadena, California, 2000. [24] F. Preparata and M. Shamos, Computational Geometry, Texts and Monographs in Computer Science, Springer-Verlag, New York, 1985. doi: 10.1007/978-1-4612-1098-6. [25] J. Ruppert, A Delaunay refinement algorithm for quality 2-dimensional mesh generation, J. Approx. Theory, 18 (1995), 548-585. doi: 10.1006/jagm.1995.1021. [26] R. Sibson, Development of strong form methods with applications in computational mechanics, in Interpolating Multivariate Data, Chapter 2 (ed. V. Barnett), John Wiley and Sons, New York, 1981. [27] H. Wendland, Error estimates for interpolation by compactly supported Radial Basis Functions of minimal degree, J. Approx. Theory, 93 (1998), 258-272. doi: 10.1006/jath.1997.3137. [28] H. Wendland, Scattered Data Approximation, Cambridge Monographs on Applied and Computational Mathematics, 17, Cambridge University Press, Cambridge, 2005. [29] X. Zhang, R. Ding and Y. Li, Adaptive RPIM meshless method, in Proceedings of the 2011 International Conference on Multimedia Technology (ICMT), IEEE, 2011, 2388-2392.
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