  We consider the stochastic Landau-Lifshitz-Gilbert equation in dimension 1. A
control process is added to the effective field. We show the existence of a
weak martingale solution for the resulting controlled equation. The proof uses
the classical Faedo-Galerkin approximation, along with the Jakubowski version
of the Skorohod Theorem. We then show pathwise uniqueness for the obtained
solution, which is then coupled with the theory of Yamada and Watanabe to give
the existence of a unique strong solution. We then show, using some semigroup
techniques that the obtained solution satisfies the maximum regularity. We then
show the existence of an optimal control. A main ingredient of the proof is
using the compact embedding of a space into itself, albeit with the weak
topology.
