  We consider the problem of overbounding and underbounding both the backward
and forward reachable set for a given polynomial vector field, nonlinear in
both state and input, with a given semialgebriac set of initial conditions and
with inputs constrained pointwise to lie in a semialgebraic set. Specifically,
we represent the forward reachable set using the value function which gives the
optimal cost to go of an optimal control problems and if smooth satisfies the
Hamilton-Jacobi- Bellman PDE. We then show that there exist polynomial upper
and lower bounds to this value function and furthermore, these polynomial
sub-value and super-value functions provide provable upper and lower bounds to
the forward reachable set. Finally, by minimizing the distance between these
sub-value and super-value functions in the L1-norm, we are able to construct
inner and outer bounds for the reachable set and show numerically on several
examples that for relatively small degree, the Hausdorff distance between these
bounds is negligible.
