Stochastic control of the Landau-Lifshitz-Gilbert equation

TL;DR

This paper studies the stochastic Landau-Lifshitz-Gilbert equation in one dimension, establishing existence, uniqueness, regularity, and optimal control of solutions using advanced probabilistic and analytical methods.

Contribution

It introduces a comprehensive framework for controlling the stochastic Landau-Lifshitz-Gilbert equation, proving existence, uniqueness, regularity, and optimal control of solutions.

Findings

Existence of a weak martingale solution.

Pathwise uniqueness and strong solution existence.

Existence of an optimal control strategy.

Abstract

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.