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Existence of piecewise affine Lyapunov functions in two dimensions

journal contribution
posted on 2023-06-07, 20:10 authored by Peter GieslPeter Giesl, Sigurdur Hafstein
In Marinosson (2002) [10], a method to compute Lyapunov functions for systems with asymptotically stable equilibria was presented. The method uses finite differences on finite elements to generate a linear programming problem for the system in question, of which every feasible solution parameterises a piecewise affine Lyapunov function. In Hafstein (2004) [2] it was proved that the method always succeeds in generating a Lyapunov function for systems with an exponentially stable equilibrium. However, the proof could not guarantee that the generated function has negative orbital derivative locally in a small neighbourhood of the equilibrium. In this article we give an example of. a system, where no piecewise affine Lyapunov function with the proposed triangulation scheme exists. This failure is due to the triangulation of the method being too coarse at the equilibrium, and we suggest a fan-like triangulation around the equilibrium. We show that for any two-dimensional system with an exponentially stable equilibrium there is a local triangulation scheme such that the system possesses a piecewise affine Lyapunov function. Hence, the method might eventually be equipped with an improved triangulation scheme that does not have deficits locally at the equilibrium.

History

Publication status

  • Published

Journal

Journal of Mathematical Analysis and Applications

ISSN

0022-247X

Publisher

Elsevier

Issue

1

Volume

371

Page range

233-248

Department affiliated with

  • Mathematics Publications

Full text available

  • No

Peer reviewed?

  • Yes

Legacy Posted Date

2012-02-06

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