In this paper, we adopt a reachability-based approach to deal with the pursuit-evasion differential game between two players in the presence of dynamic environmental disturbances (e.g., winds, sea currents). We give conditions for the game to be terminated in terms of reachable set inclusions. Level set equations are defined and solved to generate the reachable sets of the pursuer and the evader. The corresponding time-optimal trajectories and optimal strategies can be retrieved immediately afterwards. We validate our method by applying it to a pursuit-evasion game in a simple flow field, for which an analytical solution is available.We then implement the proposed scheme to a problem with a more realistic flow field.

Chinmay Kulkarni, a second year graduate student, has been selected to receive a SIAM Student Travel Award to attend the SIAM Conference on Applications of Dynamical Systems (DS17), to be held from May 21 to 25, 2017, at the Snowbird Ski and Summer Resort in Snowbird, Utah, USA.

We integrate data-driven ocean modeling with the stochastic Dynamically Orthogonal (DO) level-set optimization methodology to compute and study energy-optimal paths, speeds, and headings for ocean vehicles in the Middle-Atlantic Bight (MAB) region. We hindcast the energy-optimal paths from among exact time-optimal paths for the period 28 August 2006 to 9 September 2006. To do so, we first obtain a data-assimilative multiscale re-analysis, combining ocean observations with implicit two-way nested multiresolution primitive-equation simulations of the tidal-to-mesoscale dynamics in the region. Second, we solve the reduced-order stochastic DO level-set partial differential equations (PDEs) to compute the joint probability of minimum arrival-time, vehicle-speed time-series, and total energy utilized. Third, for each arrival time, we select the vehiclespeed time-series that minimize the total energy utilization from the marginal probability of vehicle-speed and total energy. The corresponding energy-optimal path and headings are obtained through a particle backtracking equation. Theoretically, the present methodology is PDE-based and provides fundamental energy-optimal predictions without heuristics. Computationally, it is three- to four-orders of magnitude faster than direct Monte Carlo methods. For the missions considered, we analyze the effects of the regional tidal currents, strong wind events, coastal jets, shelfbreak front, and other local circulations on the energy-optimal paths. Results showcase the opportunities for vehicles that intelligently utilize the ocean environment to minimize energy usage, rigorously integrating ocean forecasting with optimal control of autonomous vehicles.