Static PDEs for time-dependent control problems

We consider two different non-autonomous anisotropic time-optimal 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(∇u(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.