Control of harmonic map heat flow with an external field

We investigate the control problem of harmonic map heat flow by means of an external magnetic field. In contrast to the situation of a parabolic system with internal or boundary control, the magnetic field acts as the coefficients of the lower order terms of the equation. We show that for initial data whose image stays in a hemisphere, with one control acting on a subset of the domain plus a spatial-independent control acting on the whole domain, the state of the system can be steered to any ground state, i.e. any given unit vector, within any short time. To achieve this, in the first step a spatial independent control is applied to steer the solution into a small neighborhood of the peak of the hemisphere. Then under stereographic projection, the original system is reduced to an internal parabolic control system with initial data sufficiently close to $0$ such that the existing method for local controllability can be applied. The key process in this step is to give an explicit solution of an underdetermined algebraic system such that the affine type control can be converted into an internal control.