A numerical approach to a control problem in passive tracer advection by point vortex flow
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abstract
Point vortices are singular solutions of the twe>-dimensional incompressible Euler equations. These solutions correspond to the limiting case where the vorticity is completely concentrated on a finite number of spatial points each with a prescribed strength/ circulation. By definition, a passive tracer is a point vortex with zero circulation. ln our case, we consider the advection of one passive tracer by N point vortices in the unbounded
plane. ln this context, we present the formulation of certain number of control problems, as well
as the results of some numerical experiments showing the existence of optimal controls for the
cases of N = 1, N = 2, N = 3 and N = 4 vortices. More precisely, we look for the optimal
trajectories that minimize the objective function that correspond to the energy expended in the
control of the trajectories. 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
togo from the initial position zo to the final destination z1, and (iii) the maximum absolute value
Um"" that is imposed on the control variables. The latter consist in staircase controls, i.e. the
control is written as a finite linear combination of characterisc functions on the real interval.
