Optimal control of a co-rotating vortex pair: averaging and impulsive control

The problem of controlling the position of a pair of point vortices using a strain field or a field of a single source/sink is considered. The problem dimension is reduced by averaging over the fast rotation of vortices around the center of vorticity. Control is assumed to be function of rotation phase only. For the case of the strain field actuation, optimal control problem is posed in three different settings: control input whose integral over a cycle of vortex rotation is bounded, bounded control input and mixed problem where both integral over a cycle and control magnitude are bounded. It is shown that optimal solutions in integral setting are impulsive—they consist of Dirac delta functions applied at optimal phases during the cycle of vortex rotation. If a bound on the control magnitude is added to the constraints (the “mixed” problem), a solution is obtained for which control is at the maximum amplitude except possibly for intervals during which it is zero. These results are obtained by “direct” optimization. For the case of a source/sink field, we pose the optimal control problem of the averaged equations through the use of the Pontryagin maximum principle. An obtained solution is a set of Dirac delta pulses at optimal phases of the vortex rotation cycle. A discussion of these results in the context of vortex merger control is given. Impulsive control arises here naturally as an optimal control design once averaging method is used.