This work examines the controllability of planar incompressible ideal
magnetohydrodynamics (MHD). Interior controls are obtained for problems posed in
doubly-connected regions; simply-connected configurations are driven by boundary
controls. Up to now, only straight channels regulated at opposing walls have been
studied. Hence, the present program adds to the literature an exploration of
interior controllability, extends the known boundary controllability results, and
contributes ideas for treating general domains. To avoid obstacles stemming from
the MHD coupling and the magnetic field topology, a divide-and-control
strategy is proposed. This leads to a family of nonlinear velocity-controlled
subproblems which are solved using J.-M. Coron’s return method. The latter is here
developed based on a reference trajectory in the domain’s first cohomology
space.
