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Magnetization control problem for the 2D and 3D evolutionary Landau-Lifshitz-Bloch equation

Sidhartha Patnaik, Kumarasamy Sakthivel

TL;DR

The paper studies optimal control of the evolutionary Landau-Lifshitz-Bloch equation in bounded domains with external fields generated by a finite set of coils. It establishes well-posedness through strong solutions for dimensions $n\in\{1,2,3\}$ and regular solutions for $n=2$ (global) and $n=3$ (local/global under small data), and develops a rigorous optimal control framework using a stationary/adjoint approach. It derives a Fréchet-differentiable control-to-state map and a solvable adjoint system to obtain a first-order variational inequality and a projection formula for the optimal control, followed by a second-order sufficient condition on a cone of critical directions and global/uniqueness criteria. The results provide a solid mathematical basis for magnetization control in high-temperature micromagnetics, with implications for HAMR and fast switching where coil-based control is practical and constrained.

Abstract

In this study, we investigate the optimal control of the Landau-Lifshitz-Bloch equation within confined domains in $\mathbb R^n$ for $n= 2, 3.$ We establish the existence of strong solutions for dimensions $n=1, 2, 3$ under suitable growth conditions on the control, and analyze the existence and uniqueness of regular solutions. We formulate the control problem in which only a fixed set of finite magnetic field coils can constitute the external magnetic field (control). We define a cost functional by aiming at minimizing the energy discrepancy between the evolving magnetic moment and the desired state. We demonstrate the existence of an optimal solution pair and employ the classical adjoint problem approach to derive a first-order necessary optimality condition. Given the non-convex nature of the optimal control problem, we derive a second-order sufficient optimality condition using a cone of critical directions. Finally, we prove two crucial results, namely, a global optimality condition and uniqueness of an optimal control.

Magnetization control problem for the 2D and 3D evolutionary Landau-Lifshitz-Bloch equation

TL;DR

The paper studies optimal control of the evolutionary Landau-Lifshitz-Bloch equation in bounded domains with external fields generated by a finite set of coils. It establishes well-posedness through strong solutions for dimensions and regular solutions for (global) and (local/global under small data), and develops a rigorous optimal control framework using a stationary/adjoint approach. It derives a Fréchet-differentiable control-to-state map and a solvable adjoint system to obtain a first-order variational inequality and a projection formula for the optimal control, followed by a second-order sufficient condition on a cone of critical directions and global/uniqueness criteria. The results provide a solid mathematical basis for magnetization control in high-temperature micromagnetics, with implications for HAMR and fast switching where coil-based control is practical and constrained.

Abstract

In this study, we investigate the optimal control of the Landau-Lifshitz-Bloch equation within confined domains in for We establish the existence of strong solutions for dimensions under suitable growth conditions on the control, and analyze the existence and uniqueness of regular solutions. We formulate the control problem in which only a fixed set of finite magnetic field coils can constitute the external magnetic field (control). We define a cost functional by aiming at minimizing the energy discrepancy between the evolving magnetic moment and the desired state. We demonstrate the existence of an optimal solution pair and employ the classical adjoint problem approach to derive a first-order necessary optimality condition. Given the non-convex nature of the optimal control problem, we derive a second-order sufficient optimality condition using a cone of critical directions. Finally, we prove two crucial results, namely, a global optimality condition and uniqueness of an optimal control.

Paper Structure

This paper contains 17 sections, 22 theorems, 164 equations.

Table of Contents

  1. Introduction
  2. Main Results, Function Spaces and Inequalities
  3. Main Results
  4. Inequalities
  5. Strong Solution and Regular Solution
  6. Existence of Strong Solution
  7. Existence of Regular Solution
  8. Existence of Optimum and First Order Optimality Condition
  9. Existence of Optimal Control
  10. Control-to-State Operator
  11. Solvability of Adjoint System:
  12. First Order Optimality Condition
  13. Second Order Optimality Condition
  14. Control-to-Costate Operator
  15. Second Order Optimality Condition
  16. ...and 2 more sections

Key Result

Theorem 2.1

(Existence of Strong Solution for n=1,2 and 3) For any $\delta >0$, let us fix the value Suppose the control $u\in L^2(0,T;L^s(\Omega))$ and the initial data $m_0\in H^1(\Omega)$. Then there exists a strong solution $m\in \mathcal{M}_S$ of system NLP. Moreover, there exists a constant $C(\Omega,T)>0$ such that the following estimate holds: Furthermore, for $n=2$ the strong solution will be uniqu

Theorems & Definitions (37)

  • Theorem 2.1
  • Remark 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Definition 2.1: Weak formulation
  • Theorem 2.4
  • Theorem 2.5
  • Corollary 2.1
  • Definition 2.2
  • Theorem 2.6
  • ...and 27 more