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

Najla  Mohammed; Peter Giesl

doi:10.3934/dcdsb.2015.20.2453

Article Contents

2015, Volume 20, Issue 8: 2453-2476. Doi: 10.3934/dcdsb.2015.20.2453

This issue Previous Article Efficient computation of Lyapunov functions for Morse decompositions Next Article Computation of local ISS Lyapunov functions with low gains via linear programming

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

Najla Mohammed^1, and
Peter Giesl^1,

1.
Department of Mathematics, University of Sussex, Falmer BN1 9QH

Received: August 31, 2014

Revised: December 31, 2014

Published: September 2015

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Abstract

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.

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Mathematics Subject Classification: Primary: 37B25, 65N50; Secondary: 37M99, 65N35.

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