  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.
