Construction of a global Lyapunov function using radial basis functions with a single operator

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January 2007, 7(1): 101-124. doi: 10.3934/dcdsb.2007.7.101

Construction of a global Lyapunov function using radial basis functions with a single operator

1.	Zentrum Mathematik, TU München, Boltzmannstr. 3, D-85747 Garching bei München, Germany

Received April 2005 Revised August 2006 Published October 2006

Abstract
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We study the basin of attraction of an asymptotically stable equilibrium of a general autonomous ordinary differential equation. Sublevel sets of Lyapunov functions provide subsets of the basin of attraction. In this paper we construct a Lyapunov function by approximation via radial basis functions. We show the existence and the smoothness of a Lyapunov function with certain, given orbital derivative. By approximation of this Lyapunov function via its orbital derivative using radial basis functions we obtain a global Lyapunov function and can thus determine each compact subset of the basin of attraction.

Keywords: Ordinary differential equation, Lyapunov function, Zubov's equation, Basin of attraction, Radial basis functions..

Mathematics Subject Classification: Primary: 37B25, 65D05; Secondary: 34D05, 41A3.

Citation: Peter Giesl. Construction of a global Lyapunov function using radial basis functions with a single operator. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 101-124. doi: 10.3934/dcdsb.2007.7.101

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