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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-JacobiBellman 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 L<sub>1</sub>-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.

Relaxing The Hamilton Jacobi Bellman Equation To Construct Inner And Outer Bounds On Reachable Sets