Zermelo Navigation Problems on Surfaces of Revolution and Geometric Optimal Control

Bernard Bonnard (1) , Olivier Cots (2, 3) , Boris Wembe (2)

1 McTAO - Mathematics for Control, Transport and Applications
2 IRIT-APO - Algorithmes Parallèles et Optimisation
3 Toulouse INP - Institut National Polytechnique (Toulouse)

In this article, the historical study from Carathéodory-Zermelo about computing the quickest nautical path is generalized to Zermelo navigation problems on surfaces of revolution, in the frame of geometric optimal control. Using the Maximum Principle, we present two methods dedicated to analyzing the geodesic flow and to compute the conjugate and cut loci. We apply these calculations to investigate case studies related to applications in hydrodynamics, space mechanics and geometry.

Keywords: Zermelo navigation problems; Optimal control; Abnormal geodesics; Conjugate and cut loci; Regularity of the value function

Domains: Mathematics [math]

Bernard Bonnard, Olivier Cots, Boris Wembe. Zermelo Navigation Problems on Surfaces of Revolution and Geometric Optimal Control. 2023. ⟨hal-03209491v4⟩