  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
Fr\'echet differentiability of the control-to-state and control-to-costate
operators and shown the Lipschitz continuity of their Fr\'echet 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.
