Landau-Lifshitz-Gilbert equations: Controllability by Low Modes Forcing for deterministic version and Support Theorems for Stochastic version

TL;DR

This paper investigates the controllability of Landau-Lifshitz-Gilbert equations with low modes forcing, establishing controllability results for deterministic cases and support theorems for stochastic versions, advancing control theory in magnetic systems.

Contribution

It introduces a novel approach using differential geometric control theory to prove controllability of LLGEs with low mode forcing and extends results to stochastic cases with support theorems.

Findings

Global controllability of Galerkin approximations established

Approximate controllability of the full system shown

Support theorems for stochastic LLGEs derived

Abstract

In this article, we study the controllability issues of the Landau-Lifshitz-Gilbert Equations (LLGEs), accompanied with non-zero exchange energy only, in an interval in one spatial dimension with Neumann boundary conditions. The paper is of twofold. In the first part of the paper, we study the controllability issues of the LLGEs. The control force acting here is degenerate i.e., it acts through a few numbers of low mode frequencies. We exploit the Fourier series expansion of the solution. We borrow methods of differential geometric control theory (Lie bracket generating property) to establish the global controllability of the finite-dimensional Galerkin approximations of LLGEs. We show $L^2$ approximate controllability of the full system. In the second part, we consider the LLGEs with lower-dimensional degenerate random forcing (finite-dimensional Brownian motions) and study support theorems.