A Zermelo navigation problem with a vortex singularity

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, 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. 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. 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.

Abnormal geodesics in 2D-Zermelo navigation problems in the case of revolution and the fan shape of the small time balls

“…Moreover, in this situation, the time minimal value function in not continuous. Our study completes the contribution of [4] devoted to the calculation of spheres with The initial point is q 0 . The abnormal geodesic with the cusp singularity is in green while the others geodesics in red are normal.…”

“…This point of view will be developped in the next section. We refer to [4,10] for the occurence of the cusp singularity in both examples, which motivate the study of this stable singularity. The problem is set in the family of problems with rotational symmetry, which covers the two case studies.…”

“…Taking an arbitrary metric and considering the weak current case F 0 g < 1, with • g the norm associated to the metric g, this problem leads to a Zermelo navigation problem in Finsler geometry [2]. More recently, in [4], a Zermelo navigation problem was analyzed in details, associated to the evolution of a passive tracer, where the current is related to a vortex, centered at the origin of the reference frame. This problem comes at the origin from hydrodynamics [1].…”

“…It was already the case studied in details in the historical example. On the opposite, in the vortex problem [4], the analysis was concentrated on the situation where at the initial point, the current is weak but the tracer can reach the vicinity of the vortex where the current is strong.…”

Abnormal Geodesics in 2D-Zermelo Navigation Problems in the Case of Revolution and the Fan Shape of the Small Time Balls

“…It was already the case studied in details in the historical example. On the opposite, in the vortex problem [6], the analysis was concentrated on the situation where at the initial point, the current is weak, but the tracer can reach the vicinity of the vortex where the current is strong.…”