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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 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.
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https://hal.archives-ouvertes.fr/hal-02437507
Contributor : Olivier Cots Connect in order to contact the contributor
Submitted on : Monday, January 10, 2022 - 5:43:13 PM
Last modification on : Sunday, May 1, 2022 - 3:17:45 AM

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Bernard Bonnard, Olivier Cots, Joseph Gergaud, Boris Wembe. Abnormal Geodesics in 2D-Zermelo Navigation Problems in the Case of Revolution and the Fan Shape of the Small Time Balls. Systems and Control Letters, Elsevier, In press, ⟨10.1016/j.sysconle.2022.105140⟩. ⟨hal-02437507v6⟩

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