Existence of piecewise linear Lyapunov functions in arbitrary dimensions

Giesl, Peter and Hafstein, Sigurdur (2012) Existence of piecewise linear Lyapunov functions in arbitrary dimensions. Discrete and Continuous Dynamical Systems - Series A, 32 (10). pp. 3539-3565. ISSN 1078-0947

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Abstract

Lyapunov functions are an important tool to determine the basin of attraction of exponentially stable equilibria in dynamical systems. In Marinósson (2002), a method to construct Lyapunov functions was presented, using finite differences on finite elements and thus transforming the construction problem into a linear programming problem. In Hafstein (2004), it was shown that this method always succeeds in constructing a Lyapunov function, except for a small, given neighbourhood of the equilibrium.
For two-dimensional systems, this local problem was overcome by choosing a fan-like triangulation around the equilibrium. In Giesl/Hafstein (2010) the existence of a piecewise linear Lyapunov function was shown, and in Giesl/Hafstein (2012) it was shown that the above method with a fan-like triangulation always succeeds in constructing a Lyapunov function, without any local exception. However, the previous papers only considered two-dimensional systems. This paper generalises the existence of piecewise linear Lyapunov functions to arbitrary dimensions.

Item Type: Article
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Subjects: Q Science > QA Mathematics
Depositing User: Peter Giesl
Date Deposited: 30 Oct 2012 16:12
Last Modified: 07 Mar 2017 04:51
URI: http://sro.sussex.ac.uk/id/eprint/41691

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