Minimal Time Vessel Routing in a Time-Dependent Environment

We examine the two-dimensional minimal time routing problem for a vessel traveling from an origin to several ordered destination points. The sailing space is characterized by time-dependent routing properties. The controls are the power setting and the heading. For the vessel performance model, we prove that the optimal power setting always takes its upper permissible value. Moreover, appropriate first variation considerations result in local optimality conditions which, combined with global boundary conditions, form the framework of our “broken extremal” approach. The algorithmic implementation of the methodologies developed is also discussed. In particular, we emphasize that if the departure time from the origin location is known, the problem becomes much easier than the one with unspecified departure time. Elliptical bounds for the optimal state evolution are derived, significantly reducing the dimensionality of the problem. Finally, we present numerical examples based on the above methodologies.