Optimal control strategies for the Landau-Lifshitz-Gilbert equation through spatio-temporal control and fixed magnetic field coils

The magnetization control problem for the Landau-Lifshitz-Gilbert (LLG) equation mₜ= m × (Δ m +u)- m × (m × (Δ m +u)),\ (x,t) ∈ Ω× (0,T] with zero Neumann boundary data on a two-dimensional bounded domain Ω is studied when the control energy u is applied on the effective field. First, we show the existence of a weak solution, and that the magnetization vector field m satisfies an energy inequality. If a weak solution m obeys the condition that ∇ m∈ L⁴(0,T;L⁴(Ω)), then we show that it is a regular solution. The classical cost functional is modified by incorporating L⁴(0,T;L⁴(Ω))-norm of ∇ m, enabling a rigorous study of the optimal control problem. We justify the existence of an optimal control and derive first-order necessary optimality conditions using an adjoint problem approach. We establish the continuous dependency and Fr\'echet differentiability of the control-to-state and control-to-costate operators and shown the locally Lipschitz continuity of their Fr\'echet derivatives. Using these postulates, we derive a local second-order sufficient optimality condition and a global optimality criterion. Finally, we obtain another remarkable second-order sufficient optimality condition along a cone of critical directions where the control stems from a finite number of fixed magnetic field coils.