Authors:
Peter Giesl
1
;
Carlos Argáez
2
;
Sigurdur Hafstein
2
and
Holger Wendland
3
Affiliations:
1
Department of Mathematics, University of Sussex, Falmer, BN1 9QH and U.K.
;
2
Science Institute and Faculty of Physical Sciences, University of Iceland, Dunhagi 5, 107 Reykjavík and Iceland
;
3
Applied and Numerical Analysis, Department of Mathematics, University of Bayreuth, 95440 Bayreuth and Germany
Keyword(s):
Dynamical System, Complete Lyapunov Function, Quadratic Programming, Meshless Collocation.
Abstract:
A complete Lyapunov function characterizes the behaviour of a general dynamical system. In particular, the state space is split into the chain-recurrent set, where the function is constant, and the part characterizing the gradient-like flow, where the function is strictly decreasing along solutions. Moreover, the level sets of a complete Lyapunov function provide information about the stability of connected components of the chain-recurrent set and the basin of attraction of attractors therein. In a previous method, a complete Lyapunov function was constructed by approximating the solution of the PDE V0(x) = −1, where 0 denotes the orbital derivative, by meshfree collocation. We propose a new method to compute a complete Lyapunov function: we only fix the orbital derivative V0(x0) = −1 at one point, impose the constraints V0(x) ≤ 0 for all other collocation points and minimize the corresponding reproducing kernel Hilbert space norm. We show that the problem has a unique solution whic
h can be computed as the solution of a quadratic programming problem. The new method is applied to examples which show an improvement compared to previous methods.
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