Existence of Partially Quadratic Lyapunov Functions That Can Certify The Local Asymptotic Stability of Nonlinear Systems

TL;DR

This paper introduces a method using SOS programming to find partially quadratic Lyapunov functions, enabling efficient local stability certification for high-dimensional nonlinear ODEs, and proves their existence under certain conditions.

Contribution

It demonstrates the existence of partially quadratic Lyapunov functions for certifying local stability, leveraging the Center Manifold Theorem, which is novel in stability analysis.

Findings

Partially quadratic LFs can certify local stability.

The method is efficient for high-dimensional systems.

Existence of such LFs is proven under specific conditions.

Abstract

This paper proposes a method for certifying the local asymptotic stability of a given nonlinear Ordinary Differential Equation (ODE) by using Sum-of-Squares (SOS) programming to search for a partially quadratic Lyapunov Function (LF). The proposed method is particularly well suited to the stability analysis of ODEs with high dimensional state spaces. This is due to the fact that partially quadratic LFs are parametrized by fewer decision variables when compared with general SOS LFs. The main contribution of this paper is using the Center Manifold Theorem to show that partially quadratic LFs that certify the local asymptotic stability of a given ODE exist under certain conditions.