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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 atwo-dimensional bounded domain $\Omega$ is studied when the control energy $u$is applied on the effective field. First, we show the existence of a weaksolution, and the magnetization vector field $m$ satisfies an energyinequality. If a weak solution $m$ obeys the condition that $\nabla m\inL^4(0,T;L^4(\Omega)),$ then we show that it is a regular solution. Theclassical cost functional is modified by incorporating$L^4(0,T;L^4(\Omega))$-norm of $\nabla m$ so that a rigorous study of theoptimal control problem is established. Then, we justified the existence of anoptimal control and derived first-order necessary optimality conditions usingan adjoint problem approach. We have established the continuous dependency andFr\'echet differentiability of the control-to-state and control-to-costateoperators and shown the Lipschitz continuity of their Fr\'echet derivatives.Using these postulates, we derived a local second-order sufficient optimalitycondition when a control belongs to a critical cone. Finally, we also obtainanother remarkable global optimality condition posed only in terms of theadjoint state associated with the control problem.

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