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

TL;DR

This paper develops a rigorous mathematical framework for time-optimal control of the Landau-Lifshitz-Bloch equation, enabling minimal-time magnetic state manipulation across various dimensions, with implications for magnetic storage technology.

Contribution

It derives first- and second-order optimality conditions for controlling the LLB system, addressing nonlinearities and control effects in magnetic dynamics.

Findings

Existence of optimal controls for the LLB system.

First-order necessary optimality conditions derived.

Second-order sufficient conditions established.

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.