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

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 contr...

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Bibliographic Details
Published in:arXiv.org 2022-11
Main Authors: Biswas, Mrinmay, Hausenblas, Erika, Mukherjee, Debopriya
Format: Article
Language:English
Subjects:
Boundary conditions
Control theory
Controllability
Differential geometry
Fourier series
Mathematical analysis
Series expansion
Theorems
Online Access:Get full text
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Summary: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.
ISSN:2331-8422