A Simple Mathematical Model to Steering Oceanic Debris to a Targeted Region

In this article, a simplified mathematical model is presented to depict the process of collecting ocean debris. The responsible autonomous vehicles for transporting the trash are represented as passive particles, while the ocean current is simulated by the movement of point vortices on a sphere. To ensure the autonomy of the vehicles, a system of piecewise constant controls is employed, using a limited number of predetermined switching points that determine their trajectories. Each control incurs an energy cost that is aimed to be minimized. This minimization is achieved by solving a nonlinear optimization problem on the spherical surface. The initial findings indicate the existence of multiple possible trajectories for autonomous vehicles.