On the estimation of a maximal Lyapunov function and domain of attraction determination via a genetic algorithm

Abstract

The stability and the domain of attraction of autonomous nonlinear systems are important properties to be determined. This paper aims at examining a computational method for estimating the domain of attraction of nonlinear system. This very method is based on the concept of a maximal Lyapunov function candidate. The studied algorithm yields a rational Lyapunov function candidate V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sub> where its the derivative must be negative in a small neighbourhood of the equilibrium. To maximize the region of attraction, one has to minimize the numerator coefficients of V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sub> . A serious problem of nonlinear optimisation has then occurred. Our main contribution in this work is to implement a genetic algorithm technique in order to determine a maximal Lyapunov function so that we can maximize the estimate of the attraction domain of controlled nonlinear systems. The proposed approach is applied to the well known Van der Pool oscillator. Interesting results are accordingly presented in a numerical simulation study.