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

Abstract

In this article, based on two cases studies, we discuss the role of abnormal geodesics in planar Zermelo navigation problems. Such curves are limit curves of the accessibility set, in the domain where the current is strong. The problem is set in the frame of geometric time optimal control, where the control is the heading angle of the ship and in this context, abnormal curves are shown to separate time minimal curves from time maximal curves and are both small time minimizing and maximizing. We describe the small time minimal balls. For bigger time, a cusp singularity can occur in the abnormal direction, which corresponds to a conjugate point along the nonsmooth geodesic. It is interpreted in terms of regularity property of the time minimal value function.

An account on links between Finsler and Lorentz Geometries for Riemannian Geometers

“…Its elementary Cauchy boundary ∂ C M provides a completion M C and, when g R is complete, its Gromov boundary ∂ G M (see [44]) provides a compactification M G . In the case of Hadamard manifolds, this compactification agrees with the previous one by Eberlein and O'Neill (see [36]), which was introduced in a very different way by using Busemann functions associated with rays 14 . Being the main properties of these boundaries well established since the beginning of the eighties, natural questions about the relation among them, as well as its extension to (possibly non-reversible) Finslerian metrics, had remained dormant.…”

“…In the thirties of the past century this problem received the attention of some very well-known mathematicians such as Levi-Civita, Von Mises, Manià [59,66,60] and became one of the classical problems in the Calculus of Variations (see [26]). Zermelo's problem can also be solved using Optimal Control Theory (see the classical book [16] or [12,14,15,74] for recent developments), but our interest will focus on more geometrical methods, namely, the use of Finsler Geometry to solve the problem. This will be possible whenever the wind is timeindependent and its contribution to the velocity does not exceed that provided by the engine.…”

“…The systematic analysis of all these issues was carried out in [40], to be followed next. 14 Hadamard manifolds (which are complete, simply connected and with non-positive sectional curvature) become diffeomorphic to R n by Cartan-Hadamard theorem. Eberlein and O'Neill's boundary becomes a topological sphere S n−1 located at infinity.…”

Zermelo navigation problems on surfaces of revolution and geometric optimal control

“…But we extend the analysis to the case of a strong current, and we show that the cut locus splits into two branches. This phenomenon is related to the optimality status of the abnormal geodesics [15] and to the shape of the small-time balls [12]. The third case study concerns the extension of the evolution of a passive tracer near a vortex and it was analyzed in details in [13].…”

Thresholding Methods in Non-Intrusive Load Monitoring to Estimate Appliance Status

“…Zermelo himself solved this problem in R 2 and R 3 [10], and was generalized shortly after by Levi-Civita [11] to an n-dimensional plane R n and solved by means of variational calculus. Research on ZNP is still ongoing, following investigation on its generalizations [12], [13] and analysis on the effect of singularities [14], [15].…”