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IMA Journal of Applied Mathematics Advance Access originally published online on June 12, 2008
IMA Journal of Applied Mathematics 2008 73(5):782-802; doi:10.1093/imamat/hxn018
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© The Author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Construction of a local and global Lyapunov function using radial basis functions

Peter Giesl{dagger}

Department of Mathematics, University of Sussex, Mantell Building, Falmer, Brighton BN1 9RF, UK

{dagger} Email: p.a.giesl{at}sussex.ac.uk

Received on April 26, 2007; Accepted on May 2, 2008

The basin of attraction of an asymptotically stable equilibrium for an autonomous differential equation Formula can be determined through sublevel sets of a Lyapunov function. In Giesl (2007, Discrete Contin. Dyn. Syst. Ser. B, 7, 101–124), a Lyapunov function is constructed by approximating the solution of a linear partial differential equation using radial basis functions. However, the resulting Lyapunov function is non-local, i.e. it has no negative orbital derivative in a neighbourhood of the equilibrium. In this paper, we modify the construction method by using the Taylor polynomial and thus obtain a Lyapunov function with negative orbital derivative both locally and globally.

Keywords: ordinary differential equation; basin of attraction; Lyapunov function; radial basis functions.


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