Computing continuous and piecewise affine lyapunov functions for nonlinear systems

Sigurdur F. Hafstein; Christopher M. Kellett; Huijuan  Li

doi:10.3934/jcd.2015004

Article Contents

2015, Volume 2, Issue 2: 227-246. Doi: 10.3934/jcd.2015004

This issue Previous Article Variational integrators for mechanical control systems with symmetries Next Article A kernel-based method for data-driven koopman spectral analysis

Computing continuous and piecewise affine lyapunov functions for nonlinear systems

1.
School of Science and Engineering, Reykjavik University, Menntavegi 1, Reykjavik, IS-101
2.
School of Electrical Engineering and Computer Science, University of Newcastle, Callaghan, New South Wales 2308
3.
School of Mathematics and Physics, Chinese University of Geosciences (Wuhan), 430074, Wuhan

Received: May 31, 2015

Revised: February 29, 2016

Published: May 2015

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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.

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References

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