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

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.

💡 Research Summary

The paper addresses the time‑optimal control problem for the Landau‑Lifshitz‑Bloch (LLB) equation, a macroscopic model that captures magnetization dynamics of ferromagnetic materials over a broad temperature range, including the regime near and above the Curie temperature. Unlike the low‑temperature Landau‑Lifshitz‑Gilbert (LLG) model, the LLB equation incorporates longitudinal relaxation and temperature‑dependent variations of the magnetization magnitude, making it suitable for describing ultrafast processes such as heat‑assisted magnetic recording (HAMR) and femtosecond laser‑induced switching.

Mathematical setting.
The state variable is the normalized magnetization (m(t,x)\in\mathbb R^{3}) defined on a bounded domain (\Omega\subset\mathbb R^{n}) ((n=1,2,3)). After nondimensionalisation the authors consider the PDE

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