The Numerical Control of the Motion of a Passive Particle in a Point Vortex Flow

This work reports numerical explorations in the advection of one passive tracer by point vortices living in the unbounded plane. The main objective is to find the energy-optimal displacement of one passive particle (point vortex with zero circulation) surrounded by N point vortices. The direct formulation of the corresponding control problems is presented for the case of N = 1, N = 2, N = 3 and N = 4 vortices. The restrictions are due to (i) the ordinary differential equations that govern the displacement of the passive particle around the point vortices, (ii) the available time T to go from the initial position z(0) to the final destination z(f), and (iii) the maximum absolute value u(max) that is imposed on the control variables. The resulting optimization problems are solved numerically. The numerical results show the existence of nearly/quasi-optimal controls.