A Control Problem with Passive Particles Driven by Point Vortices on the Sphere

Abstract

The objective of this study is to control the motion of a passive particle advected by N point vortices in a sphere. The square of the L2 norm of control, necessary for the system to evolve from a starting point to an end point in an a priori fixed time, must be minimized. If the motion is generated by a single vortex (N= 1 ), we show that the system is controllable. The problem is also solved by a direct approach, where the control problem is transformed into a nonlinear optimization problem that is solved numerically. In the case of one (N= 1 ), two (N= 2 ), or three (N= 3 ) point vortices, the numerical results show the existence of near/quasi-optimal control.