Zermelo navigation on the sphere with revolution metrics

Bernard Bonnard (1) , Olivier Cots (2, 3) , Yannick Privat (4, 5, 6) , Emmanuel Trélat (7, 8)

1 IMB - Institut de Mathématiques de Bourgogne [Dijon]
2 IRIT-APO - Algorithmes Parallèles et Optimisation
3 IRIT - Institut de recherche en informatique de Toulouse
4 IECL - Institut Élie Cartan de Lorraine
5 SPHINX - Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization
6 IUF - Institut universitaire de France
7 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
8 CaGE - Control And GEometry

In this article motivated by physical applications, the Zermelo navigation problem on the two-dimensional sphere with a revolution metric is analyzed within the framework of minimal time optimal control. The Pontryagin maximum principle is used to compute extremal curves and a neat geometric frame is introduced using the Carathéodory-Zermelo-Goh transformation. Assuming that the current is of revolution, the geodesics are sorted according to a Morse-Reeb classification. We then illustrate the relevance of this classification using various examples from physics: the Lindblad equation in quantum control, the averaged Kepler case in space mechanics and the Landau-Lifshitz equation in ferromagnetism.

Keywords: Zermelo navigation problem; Minimal time geometric control; Morse-Reeb classification

Domains: Optimization and Control [math.OC]

Bernard Bonnard, Olivier Cots, Yannick Privat, Emmanuel Trélat. Zermelo navigation on the sphere with revolution metrics. 2024. ⟨hal-04433828⟩