A Zermelo navigation problem with a vortex singularity

TL;DR

This paper investigates a geometric optimal control problem modeling ship navigation around a vortex singularity, analyzing extremals, optimal transfer times, and the structure of optimal paths using Hamiltonian methods and numerical simulations.

Contribution

It introduces a novel analysis of a Zermelo navigation problem with vortex singularity, combining geometric control techniques with explicit computation of cut points and optimal trajectories.

Findings

Explicit characterization of extremals near vortex singularity

Computation of cut points where optimality ceases

Description of optimal spheres in weak current regions

Abstract

Helhmoltz-Kirchhoff equations of motions of vortices of an incompressible fluid in the plane define a dynamics with singularities and this leads to a Zermelo navigation problem describing the ship travel in such a field where the control is the heading angle. Considering one vortex, we define a time minimization problem which can be analyzed with the technics of geometric optimal control combined with numerical simulations, the geometric frame being the extension of Randers metrics in the punctured plane, with rotational symmetry. Candidates as minimizers are parameterized thanks to the Pontryagin Maximum Principle as extremal solutions of a Hamiltonian vector field. We analyze the time minimal solution to transfer the ship between two points where during the transfer the ship can be either in a strong current region in the vicinity of the vortex or in a weak current region. The analysis is based on a micro-local classification of the extremals using mainly the integrability properties of the dynamics due to the rotational symmetry. The discussion is complex and related to the existence of an isolated extremal (Reeb) circle due to the vortex singularity. The explicit computation of cut points where the extremal curves cease to be optimal is given and the spheres are described in the case where at the initial point the current is weak.