论文标题
时间谐波涡流方程的边界控制
Boundary control of time-harmonic eddy current equations
论文作者
论文摘要
在各种应用程序中，本文开发了频率域中麦克斯韦方程的边界控制的概念。表面卷曲被证明是适当的正则化，以使最佳控制问题得到充分。由于假定所有基础变量都具有复杂的价值，因此可与可区分性的标准结果不直接应用。取而代之的是，我们将电线衍生物的概念扩展到复杂的希尔伯特空间。最佳条件是严格得出的，并且建立了伴随变量的高阶边界规律性。使用高阶nédélec有限元元素将状态和伴随变量离散化。确定了控制控制的有限元元素空间，该空间保留了控制正则化的结构。建立了完全离散方案的融合。该理论是通过数值实验来验证的，在某些情况下是由现实应用的动机。
Motivated by various applications, this article develops the notion of boundary control for Maxwell's equations in the frequency domain. Surface curl is shown to be the appropriate regularization in order for the optimal control problem to be well-posed. Since, all underlying variables are assumed to be complex valued, the standard results on differentiability do not directly apply. Instead, we extend the notion of Wirtinger derivatives to complexified Hilbert spaces. Optimality conditions are rigorously derived and higher order boundary regularity of the adjoint variable is established. The state and adjoint variables are discretized using higher order Nédélec finite elements. The finite element space for controls is identified, as a space, which preserves the structure of the control regularization. Convergence of the fully discrete scheme is established. The theory is validated by numerical experiments, in some cases, motivated by realistic applications.
