Analysis of the magnetization control problem for the 2D evolutionary Landau-Lifshitz-Gilbert equation

Abstract

The magnetization control problem for the Landau-Lifshitz-Gilbert (LLG) equation $m_t=m\times (\Delta m+u)-m\times (m\times (\Delta m+u)),(x,t)\in \Omega \times (0,T]$ with zero Neumann boundary data on a two-dimensional bounded domain $\Omega$ is studied when the control energy $u$ is applied on the effective field. First, we show the existence of a weak solution, and the magnetization vector field $m$ satisfies an energy inequality. If a weak solution $m$ obeys the condition that $\nabla m\in L^4(0,T;L^4(\Omega )),$ then we show that it is a regular solution. The classical cost functional is modified by incorporating $L^4(0,T;L^4(\Omega ))$-norm of $\nabla m$ so that a rigorous study of the optimal control problem is established. Then, we justified the existence of an optimal control and derived first-order necessary optimality conditions using an adjoint problem approach. We have established the continuous dependency and Fréchet differentiability of the control-to-state and control-to-costate operators and shown the Lipschitz continuity of their Fréchet derivatives. Using these postulates, we derived a local second-order sufficient optimality condition when a control belongs to a critical cone. Finally, we also obtain another remarkable global optimality condition posed only in terms of the adjoint state associated with the control problem.