  This paper considers the problem of approximating the "maximal" region of
attraction (the set that contains all asymptotically stable sets) of any given
set of locally exponentially stable nonlinear Ordinary Differential Equations
(ODEs) with a sufficiently smooth vector field. Given a locally exponential
stable ODE with a differentiable vector field, we show that there exists a
globally Lipschitz continuous converse Lyapunov function whose 1-sublevel set
is equal to the maximal region of attraction of the ODE. We then propose a
sequence of d-degree Sum-of-Squares (SOS) programming problems that yields a
sequence of polynomials that converges to our proposed converse Lyapunov
function uniformly from above in the L1 norm. We show that each member of the
sequence of 1-sublevel sets of the polynomial solutions to our proposed
sequence of SOS programming problems are certifiably contained inside the
maximal region of attraction of the ODE, and moreover, we show that this
sequence of sublevel sets converges to the maximal region of attraction of the
ODE with respect to the volume metric. We provide numerical examples of
estimations of the maximal region of attraction for the Van der Pol oscillator
and a three dimensional servomechanism.
