A Converse Sum of Squares Lyapunov Result with a Degree Bound

TL;DR

This paper proves that exponential stability of polynomial vector fields on bounded sets guarantees the existence of a sum-of-squares Lyapunov function with a degree bound, enabling stability verification via semidefinite programming.

Contribution

It establishes a constructive converse Lyapunov theorem with degree bounds for polynomial systems, linking stability to SOS Lyapunov functions.

Findings

Existence of SOS Lyapunov functions for exponentially stable polynomial systems.

Degree bounds for SOS Lyapunov functions are provided.

Semidefinite programming can be used for stability analysis with complexity bounds.

Abstract

Sum of Squares programming has been used extensively over the past decade for the stability analysis of nonlinear systems but several questions remain unanswered. In this paper, we show that exponential stability of a polynomial vector field on a bounded set implies the existence of a Lyapunov function which is a sum-of-squares of polynomials. In particular, the main result states that if a system is exponentially stable on a bounded nonempty set, then there exists an SOS Lyapunov function which is exponentially decreasing on that bounded set. The proof is constructive and uses the Picard iteration. A bound on the degree of this converse Lyapunov function is also given. This result implies that semidefinite programming can be used to answer the question of stability of a polynomial vector field with a bound on complexity.