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

TL;DR

This paper investigates the role of abnormal geodesics in 2D Zermelo navigation problems, revealing their influence on optimal paths, the structure of small-time balls, and the occurrence of cusp singularities at larger times.

Contribution

It provides a detailed analysis of abnormal geodesics in planar Zermelo problems, highlighting their separating role and the geometric structure of time minimal sets.

Findings

Abnormal geodesics separate time minimal and maximal curves.

Small-time minimal balls are characterized and described.

Cusp singularities can occur at larger times, indicating conjugate points.

Abstract

In this article, based on two case 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 non-smooth image. It is interpreted in terms of the regularity property of the time minimal value function.