Time-Optimal Route Planning for Electric Ships in Stochastic Seas via PMP–HJB Equivalence

We present a framework for computing minimum-time routes for electric-propelled ships operating in stochastic sea conditions. To reflect realistic variability, we synthesize a marine environmental field using a vector autoregression (VAR) model that preserves the space–time dependencies among wind, waves, and currents. The core of the method is an equivalence-based solution that reconciles Pontryagin’s Maximum Principle (PMP) with the Hamilton–Jacobi–Bellman (HJB) equation for the time-optimal control problem. This PMP–HJB linkage enables consistent optimality with the computational efficiency of PMP and the global value-function perspective of HJB. Vessel dynamics and fluctuating sea states are incorporated directly in the trajectory computation. Robustness of the obtained routes is assessed through comprehensive parameter-sweep experiments and stress tests across ensembles of sea states. Results show that the proposed approach reliably yields minimum-time trajectories that remain stable under environmental uncertainty, providing a foundation for practical guidance toward safe and economical ship operation.