Minimum time function of a non-autonomous control system

For a time-dependent control system we consider a “reversed” minimum time problem, which consists in finding the minimum time needed by the system, whose state is initially located in a given set, to reach a given point. We show that the minimum time function constructed in this way is a unique viscosity solution of a static first order PDE, provided that, at every point of the extended phase space, admissible velocities form a convex set containing zero in the interior. We also describe a version of the Fast Marching Method (FMM) that effectively solves this PDE.