Static PDEs for time-dependent control problems

For the min-time-from-boundary problem, we show that the value function is recovered as a viscosity solution of a static Hamilton-Jacobi-Bellman partial differential equation H(ru(x),u(x),x) = 1. We demonstrate that the space-marching Ordered Upwind Methods (introduced in (29) for the autonomous control) can be extended to this non-autonomous case. We illustrate this approach with several numerical experiments. For the min-time-to-boundary problem, where no reduction to a static PDE is possible, we show how the space-marching methods can be efficiently used to approximate individual level sets of the time-dependent value function.