Stochastic Optimal Control of Finite Ensembles of Nanomagnets

We control ferromagnetic N-spin dynamics in the presence of thermal fluctuations by minimizing a quadratic functional subject to the stochastic Landau-Lifshitz-Gilbert equation. Existence of a weak solution of the stochastic optimal control problem is shown. The related first order optimality conditions consist of a coupled forward-backward SDE system, which is numerically solved by a structure-inheriting discretization, the least squares Monte-Carlo method to approximate related conditional expectations, and the new stochastic gradient method. Computational experiments are reported which motivate optimal controls in the case of interacting anisotropy, stray field, exchange energies, and acting noise.