Dynamic Programming for Finite Ensembles of Nanomagnetic Particles

We use optimal control via a distributed exterior field to steer the dynamics of an ensemble of N interacting ferromagnetic particles which are immersed into a heat bath by minimizing a quadratic functional. By using dynamic programing principle, we show the existence of a unique strong solution of the optimal control problem. By the Hopf-Cole transformation, the related Hamilton-Jacobi-Bellman equation from dynamic programming principle may be re-cast into a linear PDE on the manifold M = (S^2)^N, whose classical solution may be represented via Feynman-Kac formula. We use this probabilistic representation for Monte-Carlo simulations to illustrate optimal switching dynamics.