SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 452-475, February 2024.
Abstract. We consider space-time tracking-type distributed optimal control problems for the wave equation in the space-time domain [math], where the control is assumed to be in the energy space [math], rather than in [math], which is more common. While the latter ensures a unique state in the Sobolev space [math], this does not define a solution isomorphism. Hence, we use an appropriate state space [math] such that the wave operator becomes an isomorphism from [math] onto [math]. Using space-time finite element spaces of piecewise linear continuous basis functions on completely unstructured but shape regular simplicial meshes, we derive a priori estimates for the error [math] between the computed space-time finite element solution [math] and the target function [math] with respect to the regularization parameter [math], and the space-time finite element mesh size [math], depending on the regularity of the desired state [math]. These estimates lead to the optimal choice [math] in order to define the regularization parameter [math] for a given space-time finite element mesh size [math] or to determine the required mesh size [math] when [math] is a given constant representing the costs of the control. The theoretical results will be supported by numerical examples with targets of different regularities, including discontinuous targets. Furthermore, an adaptive space-time finite element scheme is proposed and numerically analyzed.

中文翻译：

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波动方程分布式优化控制的时空有限元方法

《SIAM 数值分析杂志》，第 62 卷，第 1 期，第 452-475 页，2024 年 2 月。
摘要。我们考虑时空域波动方程的时空跟踪型分布式最优控制问题[math]，其中假设控制在能量空间[math]中，而不是在[math]中，即比较普遍;普遍上。虽然后者确保了 Sobolev 空间 [数学] 中的唯一状态，但这并没有定义解同构。因此，我们使用适当的状态空间[math]，使得波算子成为从[math]到[math]的同构。在完全非结构化但形状规则的单纯网格上使用分段线性连续基函数的时空有限元空间，我们得出计算的时空有限元解 [math] 和目标函数 [math] 之间的误差 [math] 的先验估计。 math] 相对于正则化参数 [math] 和时空有限元网格大小 [math]，取决于所需状态 [math] 的规律性。这些估计导致最佳选择 [math]，以便为给定的时空有限元网格尺寸 [math] 定义正则化参数 [math]，或者在给定 [math] 时确定所需的网格尺寸 [math]代表控制成本的常数。理论结果将得到具有不同规律目标（包括不连续目标）的数值例子的支持。此外，提出了自适应时空有限元方案并进行了数值分析。