论文标题
时空有限元方法，用于分布式波动方程的最佳控制
Space-time finite element methods for distributed optimal control of the wave equation
论文作者
论文摘要
我们考虑时空跟踪类型在时空域中分布的最佳控制问题$ q：=ω\ times（0，t）\ subset {\ mathbb {r}}^{n+1} $，其中控制将控件假定为能量空间$ [更普遍。尽管后者确保了Sobolev空间中的唯一状态$ h^{1,1} _ {0; 0，}（q）$，但这并不能定义解决方案同构。因此，我们使用适当的状态空间$ x $，以使波运算符成为$ x $的同构，到$ [h_ {0;，0}^{1,1}（q）（q）]^*$。使用时空有限元元件空间的分段线性连续基函数在完全非结构化但形成常规的简单网格上，我们得出了错误$ \ | \ | \ | \ widetilde {u} _ {\ varrho h} _ {\ varrho h} - \ edline {rypline { $ \ widetilde {u} _ {\ varrho h} $和目标函数$ \ overline {u} $相对于正则化参数$ \ varrho $，以及时空限制元素网格网格$ h $，取决于所需的状态$ \ \ + operline的正常性。这些估计值导致最佳选择$ \ varrho = h^2 $，以定义给定时空有限元元素尺寸$ h $ $ h $ $ h $ $ \ varrho $，或确定所需的网格尺寸$ h $时，当$ \ varrho $是给定常数代表控件成本的给定常数。理论结果将由具有不同规律性（包括不连续目标）目标的数值示例支持。此外，提出了自适应时空有限元方案并进行数值分析。
We consider space-time tracking type distributed optimal control problems for the wave equation in the space-time domain $Q:= Ω\times (0,T) \subset {\mathbb{R}}^{n+1}$, where the control is assumed to be in the energy space $[H_{0;,0}^{1,1}(Q)]^*$, rather than in $L^2(Q)$ which is more common. While the latter ensures a unique state in the Sobolev space $H^{1,1}_{0;0,}(Q)$, this does not define a solution isomorphism. Hence we use an appropriate state space $X$ such that the wave operator becomes an isomorphism from $X$ onto $[H_{0;,0}^{1,1}(Q)]^*$. 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 $\|\widetilde{u}_{\varrho h}-\overline{u}\|_{L^2(Q)}$ between the computed space-time finite element solution $\widetilde{u}_{\varrho h}$ and the target function $\overline{u}$ with respect to the regularization parameter $\varrho$, and the space-time finite element mesh-size $h$, depending on the regularity of the desired state $\overline{u}$. These estimates lead to the optimal choice $\varrho=h^2$ in order to define the regularization parameter $\varrho$ for a given space-time finite element mesh size $h$, or to determine the required mesh size $h$ when $\varrho$ 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.
