Optimal Control of the 2D Landau-Lifshitz-Gilbert Equation with Control Energy in Effective Magnetic Field

TL;DR

This paper develops a mathematical framework for optimally controlling magnetization in ferromagnetic materials governed by the 2D Landau-Lifshitz-Gilbert equation, ensuring solutions exist and deriving conditions for optimal control.

Contribution

It proves existence, uniqueness, and regularity of solutions, and establishes the existence of optimal controls with necessary optimality conditions.

Findings

Global existence and uniqueness of solutions under smallness conditions.

Existence of optimal control for the system.

Derivation of first-order necessary optimality conditions.

Abstract

The optimal control of magnetization dynamics in a ferromagnetic sample at a microscopic scale is studied. The dynamics of this model is governed by the Landau-Lifshitz-Gilbert equation on a two-dimensional bounded domain with the external magnetic field (the control) applied through the effective field. We prove the global existence and uniqueness of a regular solution in $\mathbb S^2$ under a smallness condition on control and initial data. We establish the existence of optimal control and derive a first-order necessary optimality condition using the Fr\'echet derivative of the control-to-state operator and adjoint problem approach.