Construction of Lyapunov Functions Using Vector Field Decomposition

TL;DR

This paper introduces a new method for constructing Lyapunov functions through vector field decomposition, enabling stability analysis and attraction domain determination for dynamical systems.

Contribution

It proposes a novel vector field decomposition approach for Lyapunov function construction, applicable to linear and nonlinear systems, with explicit conditions and solutions.

Findings

Decomposition exists for linear systems via algebraic Riccati equation

Provides a sufficient condition for 2D systems using PDEs

Potential functions serve as Lyapunov functions under orthogonal decomposition

Abstract

In the present paper, a novel vector field decomposition based approach for constructing Lyapunov functions is proposed. For a given dynamical system, if the defining vector field admits a decomposition into two mutually orthogonal vector fields, one of which is curl-free and the other is divergence-free, then the potential function of the former can serve as a Lyapunov function candidate, since its positive definiteness will reflect the stability of the system. Moreover, under some additional conditions, its sublevel sets will give the exact attraction domain of the system. A sufficient condition for the existence of the proposed vector field decomposition is first obtained for $2$-dimensional systems by solving a partial differential equation and then generalized to $n$-dimensional systems. Furthermore, the proposed vector field decomposition always exists for linear systems and can be obtained by solving a specific algebraic Riccati equation.