Time-Extremal Navigation in Arbitrary Winds on Conformally Flat Riemannian Manifolds

Nicoleta Aldea () and Piotr Kopacz ()
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Nicoleta Aldea: Transilvania University of Braşov
Piotr Kopacz: Gdynia Maritime University

Journal of Optimization Theory and Applications, 2021, vol. 189, issue 1, No 2, 19-45

Abstract: Abstract This paper aims at solving Zermelo’s navigation problem on conformally flat Riemannian manifolds admitting a ship’s variable self-speed, under the action of arbitrary winds including space and time dependence for both perturbation and ship’s speed. Our approach is a variational one under application of the Euler–Lagrange equations with reference to the initial studies of this problem. First of all, we distinguish the navigation cases in non-critical, i.e. weak or strong, and critical winds, which are then unified into an arbitrary wind. After having considered the second variation of a given functional, we obtain the conditions for both time-minimal, i.e. the typical solutions to Zermelo’s problem, and time-maximal extremals. The anomalous paths are also emphasized. Moreover, some classification results are presented with respect to the kinds of perturbation considered separately and under an arbitrary wind. This study is illustrated at its end by a two-dimensional example including a prolate ellipsoid in the presence of a rotational vector field, wherein the solution types are being compared.

Keywords: Time-optimal; Zermelo’s navigation problem; Euler–Lagrange equations; Second variation; Conformally flat Riemannian manifold; 70G75; 70H03; 53B20; 53B50; 49J15 (search for similar items in EconPapers)
Date: 2021
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10957-021-01818-x

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