Journal of Computational Dynamics (JCD)

Computing continuous and piecewise affine lyapunov functions for nonlinear systems

Pages: 227 - 246, Volume 2, Issue 2, December 2015      doi:10.3934/jcd.2015004

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Sigurdur F. Hafstein - School of Science and Engineering, Reykjavik University, Menntavegi 1, Reykjavik, IS-101, Iceland (email)
Christopher M. Kellett - School of Electrical Engineering and Computer Science, University of Newcastle, Callaghan, New South Wales 2308, Australia (email)
Huijuan Li - School of Mathematics and Physics, Chinese University of Geosciences (Wuhan), 430074, Wuhan, China (email)

Abstract: We present a numerical technique for the computation of a Lyapunov function for nonlinear systems with an asymptotically stable equilibrium point. The proposed approach constructs a partition of the state space, called a triangulation, and then computes values at the vertices of the triangulation using a Lyapunov function from a classical converse Lyapunov theorem due to Yoshizawa. A simple interpolation of the vertex values then yields a Continuous and Piecewise Affine (CPA) function. Verification that the obtained CPA function is a Lyapunov function is shown to be equivalent to verification of several simple inequalities. Numerical examples are presented demonstrating different aspects of the proposed method.

Keywords:  Lyapunov functions, continuous and piecewise affine functions, computational techniques stability theory, ordinary differential equations.
Mathematics Subject Classification:  Primary: 93D05, 93D30, 93D20; Secondary: 93D10.

Received: June 2015;      Revised: March 2016;      Available Online: May 2016.