A Converse Sum of Squares Lyapunov Result With a Degree Bound

Although sum of squares programming has been used extensively over the past decade for the stability analysis of nonlinear systems, several fundamental 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 a sum of squares Lyapunov function which is exponentially decreasing on that bounded set. Furthermore, we derive a bound on the degree of this converse Lyapunov function as a function of the continuity and stability properties of the vector field. The proof is constructive and uses the Picard iteration. Our result implies that semidefinite programming can be used to answer the question of stability of a polynomial vector field with a bound on complexity.