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 & Continuous Dynamical Systems - B, 2015, 20 (8) : 2453-2476. doi: 10.3934/dcdsb.2015.20.2453
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show all references

References:
[1]

Springer-Verlag, Berlin, 2008. Google Scholar

[2]

in Acta Numerica, 2000, Acta Numer., 9, Cambridge Univ. Press, Cambridge, 2000, 1-38. doi: 10.1017/S0962492900000015.  Google Scholar

[3]

SIAM J. Control Optim., 40 (2001), 496-515. doi: 10.1137/S036301299936316X.  Google Scholar

[4]

in Handbook of Dynamical Systems, Vol. 2, North-Holland, Amsterdam, 2002, 221-264. doi: 10.1016/S1874-575X(02)80026-1.  Google Scholar

[5]

J. Comput. Appl. Math., 73 (1996), 65-78. doi: 10.1016/0377-0427(96)00035-0.  Google Scholar

[6]

Lecture Notes in Math., 1904, Springer, 2007.  Google Scholar

[7]

IMA J. Appl. Math., 73 (2008), 782-802. doi: 10.1093/imamat/hxn018.  Google Scholar

[8]

J. Math. Anal. Appl., 410 (2014), 292-306. doi: 10.1016/j.jmaa.2013.08.014.  Google Scholar

[9]

Discrete and Continuous Dynamical Systems - Series B, 8 (2015), 2291-2331. doi: 10.3934/dcdsb.2015.20.2291.  Google Scholar

[10]

SIAM J. Numer. Anal., 45 (2007), 1723-1741. doi: 10.1137/060658813.  Google Scholar

[11]

Numer. Math., 75 (1997), 319-337. doi: 10.1007/s002110050241.  Google Scholar

[12]

Lecture Notes in Mathematics, 1783, Springer-Verlag, Berlin, 2002. doi: 10.1007/b83677.  Google Scholar

[13]

Discrete and Continuous Dynamical Systems - Series A, 10 (2004), 657-678. doi: 10.3934/dcds.2004.10.657.  Google Scholar

[14]

Monograph. Electron. J. Diff. Eqns., (2007), 101pp. Google Scholar

[15]

Applied Mathematical Sciences, 64, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4757-3892-6.  Google Scholar

[16]

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.  Google Scholar

[17]

Springer, New York, 2014. doi: 10.1007/978-1-4419-8420-3.  Google Scholar

[18]

PhD thesis, National University of Singapore, Singapore, 2008. Google Scholar

[19]

Discrete and Continuous Dynamical Systems - Series B, 8 (2015), 2333-2360. doi: 10.3934/dcdsb.2015.20.2333.  Google Scholar

[20]

Lecture Notes in Computer Science, Springer-Verlag, Berlin, 1989. doi: 10.1007/3-540-52055-4.  Google Scholar

[21]

Ann. of Math., 50 (1949), 705-721.  Google Scholar

[22]

User's guide. Version 3.00 edition, 2013. Google Scholar

[23]

PhD thesis, California Institute of Technology Pasadena, California, 2000. Google Scholar

[24]

Texts and Monographs in Computer Science, Springer-Verlag, New York, 1985. doi: 10.1007/978-1-4612-1098-6.  Google Scholar

[25]

J. Approx. Theory, 18 (1995), 548-585. doi: 10.1006/jagm.1995.1021.  Google Scholar

[26]

in Interpolating Multivariate Data, Chapter 2 (ed. V. Barnett), John Wiley and Sons, New York, 1981. Google Scholar

[27]

J. Approx. Theory, 93 (1998), 258-272. doi: 10.1006/jath.1997.3137.  Google Scholar

[28]

Cambridge Monographs on Applied and Computational Mathematics, 17, Cambridge University Press, Cambridge, 2005.  Google Scholar

[29]

in Proceedings of the 2011 International Conference on Multimedia Technology (ICMT), IEEE, 2011, 2388-2392. Google Scholar

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