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We study the problem of steering a vehicle from its initial position in a 2D, time-varying, ocean flow field to a desired target position in minimum time. In particular, we focus on the case where the magnitude of the flow field sometimes exceeds the speed of the vehicle, and thus controllability is an issue. In order to obtain globally optimal, closed loop trajectories, one solves a dynamic Hamilton Jacobi Bellman equation for the optimal “time-to-go” and associated optimal feedback control law. We do this indirectly via a simple but powerful extremal field algorithm, which allows incremental refinement of the solution and is trivial to parallelize. We characterize solutions and the resulting closed loop optimal trajectories for a time-invariant double gyre flow field and for a numerically-defined, time-varying flow field from a real model of the Adriatic Sea.