  We propose a novel formulation for approximating reachable sets through a
minimum discounted reward optimal control problem. The formulation yields a
continuous solution that can be obtained by solving a Hamilton-Jacobi equation.
Furthermore, the numerical approximation to this solution can be obtained as
the unique fixed-point to a contraction mapping. This allows for more efficient
solution methods that could not be applied under traditional formulations for
solving reachable sets. In addition, this formulation provides a link between
reinforcement learning and learning reachable sets for systems with unknown
dynamics, allowing algorithms from the former to be applied to the latter. We
use two benchmark examples, double integrator, and pursuit-evasion games, to
show the correctness of the formulation as well as its strengths in comparison
to previous work.
