论文标题
抛物线分布式最佳控制问题的强大时空有限元误差估计与能量正则化的最佳控制问题
Robust space-time finite element error estimates for parabolic distributed optimal control problems with energy regularization
论文作者
论文摘要
我们考虑时空跟踪线性para \ -bo \ -bo \ -bo \ -bo \ -bo \ -bo \ - 时空气缸中给出的初始边界价值问题，$ q =ω\ times（0，t）$，并且由右侧$ z_ \ varrho $控制的bochner space $ l^2（0，t; h^2（0，t;因此，自然可以用$ l^2（0，t; h^{ - 1}（ω））$ norm替换常规$ l^2（q）$ norm正则化。我们得出了错误$ \ | \ widetilde {u} _ {\ varrho h} - \ bar { $ \ varrho $和时空有限元网格大小$ h $，并取决于所需状态$ \ bar {u} $的规律性。这些估计值导致最佳选择$ \ varrho = h^2 $。近似状态$ \ wideTilde {u} _ {\ varrho h} $是通过时空有限元元素方法使用分段线性和连续的基础函数来计算的，该函数在完全非结构化的简单网格上，以$ q $。理论结果通过两个空间维度和三个空间维度的一系列数值示例进行了定量说明。
We consider space-time tracking optimal control problems for linear para\-bo\-lic initial boundary value problems that are given in the space-time cylinder $Q = Ω\times (0,T)$, and that are controlled by the right-hand side $z_\varrho$ from the Bochner space $L^2(0,T;H^{-1}(Ω))$. So it is natural to replace the usual $L^2(Q)$ norm regularization by the energy regularization in the $L^2(0,T;H^{-1}(Ω))$ norm. We derive a priori estimates for the error $\|\widetilde{u}_{\varrho h} - \bar{u}\|_{L^2(Q)}$ between the computed state $\widetilde{u}_{\varrho h}$ and the desired state $\bar{u}$ 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 $\bar{u}$. These 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.
