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 × (Δm + u) − m × (m × (Δm + 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. Then, 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 Fréchet differentiability of the control-to-state and control-to-costate operators and show the locally Lipschitz continuity of their Frechet derivatives. Using these postulates, we derive a local second-order sufficient optimality condition. Moreover, we modified the control problem and derived 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.