As in our previous work (SINUM 59(2):660–674, 2021) we consider space-time tracking optimal control problems for linear parabolic initial boundary value problems that are given in the space-time cylinder \(Q = \Omega \times (0,T)\), and that are controlled by the right-hand side \(z_\varrho \) from the Bochner space \(L^2(0,T;H^{-1}(\Omega ))\). So it is natural to replace the usual \(L^2(Q)\) norm regularization by the energy regularization in the \(L^2(0,T;H^{-1}(\Omega ))\) norm. We derive new a priori estimates for the error \(\Vert \widetilde{u}_{\varrho h} - \overline{u}\Vert _{L^2(Q)}\) between the computed state \(\widetilde{u}_{\varrho h}\) and the desired state \(\overline{u}\) in terms of the regularization parameter \(\varrho \) and the space-time finite element mesh size h, and depending on the regularity of the desired state \(\overline{u}\). These new estimates lead to the optimal choice \(\varrho = h^2\). The approximate state \(\widetilde{u}_{\varrho h}\) is computed by means of a space-time finite element method using piecewise linear and continuous basis functions on completely unstructured simplicial meshes for Q. The theoretical results are quantitatively illustrated by a series of numerical examples in two and three space dimensions. We also provide performance studies for different solvers.

正如我们之前的工作（SINUM 59(2):660–674, 2021）一样，我们考虑线性抛物线初始边值问题的时空跟踪最优控制问题，这些问题在时空柱面\(Q = \Omega \次 (0,T)\) ，并且由博赫纳空间\(L^2(0,T;H^{-1}(\Omega ) )的右侧\(z_\varrho \)控制)\)。所以很自然地用\(L^2(0,T;H^{-1}(\Omega ))\)范数中的能量正则化代替通常的\(L^2(Q)\)范数正则化。我们对计算状态\( \根据正则化参数\(\varrho \)和时空有限元网格大小h ，以及取决于期望状态的规律性\(\overline{u}\)。这些新的估计导致最优选择\(\varrho = h^2\)。近似状态\(\widetilde{u}_{\varrho h}\)是通过时空有限元方法计算的，在Q的完全非结构化单纯网格上使用分段线性和连续基函数。通过二维和三维空间中的一系列数值例子定量地说明了理论结果。我们还提供不同求解器的性能研究。