Modified refinement algorithm to construct Lyapunov functions using meshless collocation. (English) Zbl 1515.37097
Summary: Lyapunov functions are functions with negative derivative along solutions of a given ordinary differential equation. Moreover, sublevel sets of a Lyapunov function are subsets of the domain of attraction of the equilibrium. One of the numerical construction methods for Lyapunov functions uses meshless collocation with radial basis functions.
Recently, this method was combined with a grid refinement algorithm (GRA) to reduce the number of collocation points needed to construct Lyapunov functions. However, depending on the choice of the initial set of collocation point, the algorithm can terminate, failing to compute a Lyapunov function. In this paper, we propose a modified grid refinement algorithm (MGRA), which overcomes these shortcomings by adding appropriate collocation points using a clustering algorithm. The modified algorithm is applied to two- and three-dimensional examples.
Recently, this method was combined with a grid refinement algorithm (GRA) to reduce the number of collocation points needed to construct Lyapunov functions. However, depending on the choice of the initial set of collocation point, the algorithm can terminate, failing to compute a Lyapunov function. In this paper, we propose a modified grid refinement algorithm (MGRA), which overcomes these shortcomings by adding appropriate collocation points using a clustering algorithm. The modified algorithm is applied to two- and three-dimensional examples.
MSC:
| 37M21 | Computational methods for invariant manifolds of dynamical systems |
| 37M22 | Computational methods for attractors of dynamical systems |
| 65P40 | Numerical nonlinear stabilities in dynamical systems |
Keywords:
differential equation; Lyapunov function; basin of attraction; meshfree collocation; refinementReferences:
| [1] | C. C. C. K. Aggarwal Reddy, Data Clustering: Algorithms and Applications (2014) · Zbl 1331.62026 · doi:10.1201/b15410 |
| [2] | M. de Berg, O. Cheong, M. van Kerveld and M. Overmars, Computational geometry: Algorithms and Applications, Springer-Verlag, Berlin, 2008. · Zbl 1140.68069 |
| [3] | M. D. Buhmann, Radial Basis Functions: Theory and Implementations, volume 12 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2003. · Zbl 1038.41001 |
| [4] | S. L. Chiu, Fuzzy model identification based on cluster estimation, Journal of Intelligent & Fuzzy Systems, 2, 267-278 (1994) · doi:10.3233/IFS-1994-2306 |
| [5] | P. Giesl, Construction of Global Lyapunov Functions Using Radial Basis Functions, Lecture Notes in Math. 1904, Springer, 2007. |
| [6] | P. Giesl, Construction of a local and global Lyapunov function using radial basis functions, IMA J. Appl. Math., 73, 782-802 (2008) · Zbl 1155.34030 · doi:10.1093/imamat/hxn018 |
| [7] | P. S. Giesl Hafstein, Review on computational methods for Lyapunov functions, Discrete Contin. Dyn. Syst. Ser. B, 20, 2291-2331 (2015) · Zbl 1337.37001 · doi:10.3934/dcdsb.2015.20.2291 |
| [8] | P. Giesl and N. Mohammed, Combination of refinement and verification for the construction of Lyapunov functions using radial basis functions, In Proceedings of the 15th International Conference on Informatics in Control, Automation and Robotics (ICINCO), IEEE, 2018,569-578. · Zbl 1530.93436 |
| [9] | P. N. Giesl Mohammed, Verification estimates for the construction of Lyapunov functions using meshfree collocation, Discrete Contin. Dyn. Syst. Ser. B, 24, 4955-4981 (2019) · Zbl 1462.65093 · doi:10.3934/dcdsb.2019040 |
| [10] | P. H. Giesl Wendland, Meshless collocation: Error estimates with application to Dynamical Systems, SIAM J. Numer. Anal., 45, 1723-1741 (2007) · Zbl 1148.65090 · doi:10.1137/060658813 |
| [11] | S. F. Hafstein, A constructive converse Lyapunov theorem on exponential stability, Discrete and Continuous Dynamical Systems - Series A, 10, 657-678 (2004) · Zbl 1070.93044 · doi:10.3934/dcds.2004.10.657 |
| [12] | K. Hammouda and F Karray, A comparative study on data clustering techniques, University of Waterloo, Ontario, Canada, 2000. |
| [13] | S. S. Iyengar, K. G. Boroojeni and N. Balakrishnan, Mathematical Theories of Distributed Sensor Networks, Springer, New York, 2014. · Zbl 1301.68004 |
| [14] | T. A. Johansen, Computation of Lyapunov functions for smooth, nonlinear systems using convex optimization, Automatica, 36, 1617-1626 (2000) · Zbl 0971.93069 · doi:10.1016/S0005-1098(00)00088-1 |
| [15] | C. M. Kellett, Classical converse theorems in Lyapunov’s second method, Discrete Contin. Dyn. Syst. Ser. B, 20, 2333-2360 (2015) · Zbl 1332.93292 · doi:10.3934/dcdsb.2015.20.2333 |
| [16] | R. Klein, Concrete and Abstract Voronoi Diagrams, Lecture Notes in Computer Science. Springer-Verlag, Berlin, 1989. · Zbl 0699.68005 |
| [17] | J. A. J. D. Leskovec Rajaraman Ullman, Mining of Massive DataSets (2014) · doi:10.1017/CBO9781139924801 |
| [18] | J. L. Massera, On Liapounoff’s conditions of stability, Ann. of Math., 50, 705-721 (1949) · Zbl 0038.25003 · doi:10.2307/1969558 |
| [19] | V. Moertini, Introduction to five data clustering algorithms, INTEGRAL, 7 (2002). |
| [20] | N. Mohammed, Grid Refinement and Verification Estimates for the RBF Construction Method of Lyapunov Functions, Doctoral thesis (PhD), University of Sussex, 2016. |
| [21] | N. P. Mohammed Giesl, Grid refinement in the construction of Lyapunov functions using radial basis functions, Discrete Contin. Dyn. Syst. Ser. B, 20, 2453-2476 (2015) · Zbl 1366.37147 · doi:10.3934/dcdsb.2015.20.2453 |
| [22] | P. Parrilo, Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimiza, PhD thesis: California Institute of Technology Pasadena, California, 2000. |
| [23] | P. K. Pisharady, P. Vadakkepat and L. A. Poh, Computational Intelligence in Multi-feature Visual Pattern Recognition. Hand Posture and Face Recognition using Biological Inspired Approaches, Springer Science+Business Media, Singapore, 2014. |
| [24] | M. J. D. Powell, The theory of radial basis function approximation in 1990, In Advances in Numerical Analysis, Vol. II (Lancaster, 1990), Oxford Sci. Publ., Oxford Univ. Press, New York, 1992,105-210. · Zbl 0787.65005 |
| [25] | F. P. Preparata and M. I. Shamos, Computational Geometry, Texts and Monographs in Computer Science, Springer-Verlag, New York, 1985. · Zbl 0575.68059 |
| [26] | R. H. Schaback Wendland, Kernel techniques: From machine learning to meshless methods, Acta Numer., 15, 543-639 (2006) · Zbl 1111.65108 · doi:10.1017/S0962492906270016 |
| [27] | V. Torra, Information Fusion in Data Mining, Springer-Verlag, Berlin, 2003. · Zbl 1030.68100 |
| [28] | H. Wendland, Error estimates for interpolation by compactly supported Radial Basis Functions of minimal degree, J. Approx. Theory, 93, 258-272 (1998) · Zbl 0904.41013 · doi:10.1006/jath.1997.3137 |
| [29] | H. Wendland, Scattered Data Approximation, volume 17 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2005. · Zbl 1075.65021 |
| [30] | R. R. D. P. Yager Filev, Generation of fuzzy rules by mountain clustering, Journal of Intelligent & Fuzzy Systems, 2, 209-219 (1993) · doi:10.3233/IFS-1994-2301 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
