Optimal control in evolutionary micromagnetism
Optimal control in evolutionary micromagnetism
复制标题
演化微磁学中的最优控制
DOI：
10.1093/imanum/dru034
复制
发表时间：
2015
影响因子：
2.1
通讯作者：
A. Schäfer
中科院分区：
文献类型：
--
作者：
T. Dunst;M. Klein;A. Prohl;A. Schäfer
我们考虑服从一维 Landau-Lifshitz-Gilbert 方程的最优控制问题，该方程描述了磁化强度的演化。该问题的提出是为了控制铁磁体的切换过程。推导了最优系统的存在性和一阶必要最优系统。我们展示了时间步长消失的时间离散化（半隐式欧拉方法）的状态、伴随和控制变量（直到子序列）的收敛性。这里的一个主要步骤是验证半离散状态的相应稳定性属性，这很重要，因为迭代采用的值仅接近 2。我们在变分离散化中使用扰动参数，以便显示半离散状态变量的误差界限，然后我们可以从中推断出半离散状态以及伴随变量的统一界限。数值研究强调了这些结果，并将这种离散化与进一步的变体进行比较，该变体基于状态方程的投影策略，以增强迭代以更好地近似2。
We consider an optimal control problem subject to the one-dimensional Landau–Lifshitz–Gilbert equation, which describes the evolution of magnetizations in2. 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 semidiscrete state, which is nontrivial since the iterates take values which only approximate2. We use a perturbation argument within a variational discretization in order to show error bounds for the semidiscrete state variables, from which we may then infer uniform bounds for the semidiscrete 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 approximate2.