Realizability of vortex permutations with stationary boundary control

Determine whether permutations of vortex endpoints are realizable by stationary localized Dirichlet or compatible Neumann boundary controls for the two-dimensional point-vortex system, including exchanges in which the prescribed trajectories cannot be represented by mutually disjoint arcs.

Background

The stationary controllability theorem constructs controls whose vortex trajectories approximate prescribed mutually disjoint real-analytic arcs and exactly connect the specified initial and final configurations. The disjointness assumption excludes endpoint permutations, such as exchanging two vortices. Because the stationary harmonic field is constructed separately near each prescribed arc, the proof does not address trajectories involving such exchanges, leaving their realizability unresolved.