Gradual approximation of the domain of attraction by gradual extension of the "embryo" of the transformed optimal Lyapunov function

Abstract

In this paper an autonomous analytical system of ordinary differential equations is considered. For an asymptotically stable steady state $x^0$ of the system a gradual approximation of the domain of attraction (DA) is presented in the case when the matrix of the linearized system in $x^0$ is diagonalizable. This technique is based on the gradual extension of the "embryo" of an analytic function of several complex variables. The analytic function is the transformed of a Lyapunov function whose natural domain of analyticity is the DA and which satisfies a linear non-homogeneous partial differential equation. The equation permits to establish an "embryo" of the transformed function and a first approximation of DA. The "embryo" is used for the determination of a new "embryo" and a new part of the DA. In this way, computing new "embryos" and new domains, the DA is gradually approximated. Numerical examples are given for polynomial systems. For systems considered recently in the literature the results are compared with those obtained with other methods.