Dynamic Programming for Finite Ensembles of Nanomagnetic Particles

TL;DR

This paper develops a control framework for steering the behavior of nanomagnetic particle ensembles immersed in heat baths, using dynamic programming and probabilistic methods to find optimal switching strategies.

Contribution

It introduces a novel optimal control approach for ferromagnetic particle ensembles, employing dynamic programming and probabilistic PDE solutions.

Findings

Existence of a unique strong solution to the control problem.

Representation of the Hamilton-Jacobi-Bellman equation as a linear PDE.

Monte-Carlo simulations demonstrating optimal switching dynamics.

Abstract

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.