The Zermelo Navigation Problem on the 2-Sphere of Revolution: An Optimal Control Perspective with Applications to Micromagnetism

This article explores recent geometric optimal control techniques for the analysis of geodesics in Zermelo navigation problems on 2-spheres of revolution, focusing on accessibility and optimality properties. These techniques involve the classification of pairs \((F_0, g)\), where \(F_0\) represents the current and \(g\) is the Riemannian metric of revolution on the 2-sphere. By applying the maximum principle, the geodesic dynamics are described by a Hamiltonian vector field on the cotangent bundle \(T^{*}S^{2}\), which remains invariant under positive homothety in the fiber. The primary motivation of this study is to investigate the application of these techniques to micromagnetism, particularly in the context of spin magnetization reversal. The underlying model is complex, depending on four parameters as well as the control amplitude of the applied magnetic field. The analysis is further supported by algebraic geometry and numerical simulations.