  In this paper, we propose an iterative method for using SOS programming to
estimate the region of attraction of a polynomial vector field, the conjectured
convergence of which necessitates the existence of polynomial Lyapunov
functions whose sublevel sets approximate the true region of attraction
arbitrarily well. The main technical result of the paper is the proof of
existence of such a Lyapunov function. Specifically, we use the Hausdorff
distance metric to analyze convergence and in the main theorem demonstrate that
the existence of an $n$-times continuously differentiable maximal Lyapunov
function implies that for any $\epsilon>0$, there exists a polynomial Lyapunov
function and associated sub-level set which together prove stability of a set
which is within $\epsilon$ Hausdorff distance of the true region of attraction.
The proposed iterative method and probably convergence is illustrated with a
numerical example.
