Time-optimal navigation in arbitrary winds. (2020) - Aldea, Nicoleta
Abstract: The paper presents an improved formula for optimal navigation with respect to time which can be applied to the practical methods and numerical techniques for computing minimum-time paths and routing in arbitary winds, i.e. strong, critical, weak including their combinations, considered on arbitrary modelling surface, thought of as a Riemannian manifold of dimension two. It admits space and time dependence of both, a perturbing wind and a self-speed (airspeed) of a ship. The generalized formula stands for the refinement of analogous optimality conditions worked out recently for spherical cases and so, the improvement of the computational aspects of the corresponding algorithms. In particular, the findings can lead to extension of previous models to ellipsoidal with a two-state problem and two controls. The results can be used for air and marine applications which refer to the Zermelo navigation problem. Although the main interest is dedicated to least time solutions, our objective is also to point out and compare the second type of optimal trajectories, i.e. time-maximal in strong perturbation, where speed of a wind (a water current or a stream) is greater than maximum speed of a ship (an aerial vehicle) relative to a flowing water (air). We also obtain the classification results involving time-minimal and time-maximal paths, types of winds and optimal controls. For the purposes of applied modelling an improved optimality condition in the spherical coordinates for time-optimal navigation on a rotational ellipsoid is worked out in the presented example.