Optimal control in evolutionary micromagnetism

Abstract

Abstract. We consider an optimal control problem subject to the Landau-Lifshitz-Gilbert equation which describes the evolution of magnetizations in S 2 . The problem is motivated in order to control switching processes of ferromagnets. Existence of an optimum and the first order necessary optimality system are derived. We show (up to subsequences) convergence of state, adjoint and control variables of a time discretization (semi-implicit Euler method) for vanishing time step size. A main step here is to verify corresponding stability properties for the semi-discrete state, which is nontrivial since the iterates take values which only approximate S 2 . We use a perturbation argument within a variational discretization in order to show error bounds for the semi-discrete state variables, from which we may then infer uniform bounds for the semi-discrete state and also adjoint variables. Numerical studies underline these results and compare this discretization with a further variant which bases on a projection strategy for the state equation to enhance iterates to better approximate S 2 .

Weak Solutions of a Stochastic Landau-Lifshitz-Gilbert Equation

“…As indicated earlier, our aim is to try and optimize the switching of magnetization by giving some external input. Following the works [28], [29], we add the control to the effective field. We now describe the inclusion of the control process in the stochastic LLG equation.…”

“…The initial studies for deterministic optimal control of ferromagnetic dynamics were done in [2], [3]. Dunst, Klein, Prohl and Schäfer in [28] show the existence of an optimal control subject to a one dimensional Landau-Lifshitz-Gilbert equation. Dunst, Majee, Prohl and Vallet in [29] consider the stochastic counterpart of the above problem.…”

Wong–Zakai approximations for the stochastic Landau–Lifshitz–Bloch equations

“…Following the lead of [16] (where a control problem for the LLG equation in dimension 1 is studied), Dunst et. al.…”

Optimal control of the 2D Landau-Lifshitz-Gilbert equation with control energy in effective magnetic field

“…The current paper discusses the optimal control of magnetization dynamics in ferromagnetic material governed by the Landau-Lifshitz-Gilbert (LLG) equation. The control problem has numerous physical applications, including magnetic sensors and data storage devices( [13], [18], [24], [28]). In these applications, it is of utter importance that we would precisely control the magnetization process with the help of some applied magnetic field.…”