Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Existence of piecewise linear Lyapunov functions in arbitrary dimensions

Pages: 3539 - 3565, Volume 32, Issue 10, October 2012      doi:10.3934/dcds.2012.32.3539

       Abstract        References        Full Text (1016.8K)       Related Articles       

Peter Giesl - Department of Mathematics, University of Sussex, Falmer BN1 9QH, United Kingdom (email)
Sigurdur Hafstein - School of Science and Engineering, Reykjavik University, Menntavegi 1, IS-101 Reykjavik, Iceland (email)

Abstract: Lyapunov functions are an important tool to determine the basin of attraction of exponentially stable equilibria in dynamical systems. In MarinĂ³sson (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.

Keywords:  Lyapunov function, existence, piecewise linear, exponentially stable equilibrium, triangulation.
Mathematics Subject Classification:  Primary: 37B25; Secondary: 34D20, 37C75.

Received: May 2011;      Revised: October 2011;      Available Online: May 2012.