The Zermelo navigation problem on the 2-sphere of revolution: an optimal control perspective with applications to micromagnetism

Abstract This article presents geometric optimal control techniques for analyzing geodesics in time-optimal Zermelo navigation problems on 2-spheres of revolution. We classify the problem by analyzing the pair ( F 0 , g ) , which represents the current (or wind) and the Riemannian metric. Using the maximum principle, the dynamics of geodesics are described by a Hamiltonian vector field on the cotangent bundle T ∗ S 2 . Our primary motivation is the application to micromagnetism, specifically spin magnetization reversal in ferromagnetic ellipsoidal samples. This model depends on four parameters and the amplitude of the applied magnetic field. The problem is formulated as a Zermelo navigation on the 2-sphere, where geodesics are classified as elliptic, hyperbolic, or abnormal. We demonstrate that the transition set | F 0 | g = 1 , which separates weak and strong current domains, is critical for understanding optimality. A key result shows that abnormal geodesics intersect this set with semi-cubical cusp singularities, a phenomenon we term the Landau–Lifshitz billiard. The analysis of the transition set’s connected components is complex and complemented by algebraic geometry and symbolic computations. We further reveal that hyperbolic geodesics lose optimality at their second intersection with the abnormal arc. Our numerical simulations complement this analysis by computing conjugate and cut loci, wavefronts, and accessibility sets, providing new insights into optimal magnetization switching under bounded control.