Relaxed optimal control for the stochastic Landau-Lifshitz-Gilbert equation

TL;DR

This paper develops a relaxed optimal control framework for the stochastic Landau-Lifshitz-Gilbert equation, enabling the determination of optimal controls in complex, non-convex settings involving thermal fluctuations and non-linear dependencies.

Contribution

It introduces a relaxed control approach using Young measures for the stochastic LLG equation with general cost functionals and non-linear control dependence.

Findings

Existence of an optimal relaxed control for the stochastic LLG equation.

Application of Young measures and Skorohod Theorem for control relaxation.

Framework accommodates non-convex costs and non-linear control dependencies.

Abstract

We consider the stochastic Landau-Lifshitz-Gilbert equation, perturbed by a real-valued Wiener process. We add an external control to the effective field as an attempt to drive the magnetization to a desired state and also to control thermal fluctuations. We use the theory of Young measures to relax the given control problem along with the associated cost. We consider a control operator that can depend (possibly non-linearly) on both the control and the associated solution. Moreover, we consider a fairly general associated cost functional without any special convexity assumption. We use certain compactness arguments, along with the Jakubowski version of the Skorohod Theorem to show that the relaxed problem admits an optimal control.