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Time Optimal Control Problem for the Landau-Lifshitz-Bloch equation

Sidhartha Patnaik, Kumarasamy Sakthivel

TL;DR

This work develops a rigorous time-optimal control framework for the Landau-Lifshitz-Bloch equation on bounded domains in 1–3 dimensions, addressing the nonlinear diffusion-relaxation dynamics that arise at elevated temperatures. It establishes the existence of regular solutions and a time-optimal control, and derives first-order necessary conditions via an adjoint system alongside a second-order sufficient condition for local optimality. The analysis handles challenging nonlinearities such as $m\times\Delta m$ and $(1+|m|^2)m$ and the nonlinearly appearing control, by constructing a unified solvability theory for linearized and second-variation systems. These results provide a solid theoretical foundation for rapid magnetic state manipulation and lay groundwork for efficient, adjoint-based numerical schemes in ultrafast spin dynamics and thermally assisted switching.

Abstract

This paper investigates the time-optimal control problem for the Landau-Lifshitz-Bloch (LLB) equation, a macroscopic model that characterizes magnetization dynamics in ferromagnetic materials across a wide temperature range, including near and above the Curie temperature. We analyze the LLB system on bounded domains in one, two, and three dimensions, establishing the existence of optimal controls that drive the magnetization to a desired target state within a minimal time frame. Utilizing a Lagrange multiplier approach and an adjoint-based framework, we derive first-order necessary optimality conditions. Furthermore, we establish second-order sufficient conditions for local optimality, addressing the mathematical challenges posed by the system's inherent nonlinearities and the nonlinear appearance of the control in the effective magnetic field. These results provide a rigorous theoretical basis for the rapid manipulation of magnetic states, offering insights into the fundamental limits of control for nonlinear diffusion-relaxation processes in magnetism. Such findings are essential for advancing high-speed magnetic memory technologies and optimizing thermal magnetic switching in next-generation storage technologies.

Time Optimal Control Problem for the Landau-Lifshitz-Bloch equation

TL;DR

This work develops a rigorous time-optimal control framework for the Landau-Lifshitz-Bloch equation on bounded domains in 1–3 dimensions, addressing the nonlinear diffusion-relaxation dynamics that arise at elevated temperatures. It establishes the existence of regular solutions and a time-optimal control, and derives first-order necessary conditions via an adjoint system alongside a second-order sufficient condition for local optimality. The analysis handles challenging nonlinearities such as and and the nonlinearly appearing control, by constructing a unified solvability theory for linearized and second-variation systems. These results provide a solid theoretical foundation for rapid magnetic state manipulation and lay groundwork for efficient, adjoint-based numerical schemes in ultrafast spin dynamics and thermally assisted switching.

Abstract

This paper investigates the time-optimal control problem for the Landau-Lifshitz-Bloch (LLB) equation, a macroscopic model that characterizes magnetization dynamics in ferromagnetic materials across a wide temperature range, including near and above the Curie temperature. We analyze the LLB system on bounded domains in one, two, and three dimensions, establishing the existence of optimal controls that drive the magnetization to a desired target state within a minimal time frame. Utilizing a Lagrange multiplier approach and an adjoint-based framework, we derive first-order necessary optimality conditions. Furthermore, we establish second-order sufficient conditions for local optimality, addressing the mathematical challenges posed by the system's inherent nonlinearities and the nonlinear appearance of the control in the effective magnetic field. These results provide a rigorous theoretical basis for the rapid manipulation of magnetic states, offering insights into the fundamental limits of control for nonlinear diffusion-relaxation processes in magnetism. Such findings are essential for advancing high-speed magnetic memory technologies and optimizing thermal magnetic switching in next-generation storage technologies.

Paper Structure

This paper contains 11 sections, 16 theorems, 170 equations.

Table of Contents

  1. Introduction
  2. Main Results and Inequalities
  3. Main Results
  4. Inequalities
  5. Existence of Optimal Control
  6. Linearized System and First-Order Optimality Condition
  7. Linearized System
  8. First-Order Optimality Condition
  9. Second-Order Optimality Condition
  10. Conclusion and Future Works
  11. Appendix

Key Result

Theorem 2.1

(Existence of Regular Solution for n = 1,2,3) Let $u\in L^2(0,T;H^1(\Omega))$ be the control and $m_0$ be the initial data satisfying condition IC. For $n=2$, the system NLP admits a unique global regular solution $m\in \mathcal{M}$. Moreover, there exist constants $M(m_0,u,\Omega,T,n)$ and $C(\Omeg For $n=3$, the system NLP admits a unique "local in time" regular solution for every control in $L^

Theorems & Definitions (26)

  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.1
  • Proposition 2.1
  • Definition 2.1: Weak formulation
  • Theorem 2.3
  • Theorem 2.4: First-Order Optimality Condition
  • Theorem 2.5: Second-Order Optimality Condition
  • Lemma 2.1
  • Lemma 2.2
  • ...and 16 more