Exploring Controlled Passive Particle Motion Driven by Point Vortices on a Sphere

Abstract

This work focuses on optimizing the displacement of a passive particle interacting with vortices located on the surface of a sphere. The goal is to minimize the energy expended during the displacement within a fixed time. The modeling of particle dynamics, whether in Cartesian or spherical coordinates, gives rise to alternative formulations of the identical problem. Thanks to these two versions of the same problem, we can assert that the algorithm, employed to transform the optimal control problem into an optimization problem, is effective, as evidenced by the obtained controls. The numerical resolution of these formulations through a direct approach consistently produces optimal solutions, regardless of the selected coordinate system.