We consider a distributed optimal control problem subject to a parabolic evolution equation as constraint. The control will be considered in the energy norm of the anisotropic Sobolev space $[H_{0;,0}^{1,1/2}(Q)]^\ast$, such that the state equation of the partial differential equation defines an isomorphism onto $H^{1,1/2}_{0;0,}(Q)$. Thus, we can eliminate the control from the tracking type functional to be minimized, to derive the optimality system in order to determine the state. Since the appearing operator induces an equivalent norm in $H_{0;0,}^{1,1/2}(Q)$, we will replace it by a computable realization of the anisotropic Sobolev norm, using a modified Hilbert transformation. We are then able to link the cost or regularization parameter $\varrho>0$ to the distance of the state and the desired target, solely depending on the regularity of the target. For a conforming space-time finite element discretization, this behavior carries over to the discrete setting, leading to an optimal choice $\varrho = h_x^2$ of the regularization parameter $\varrho$ to the spatial finite element mesh size $h_x$. Using a space-time tensor product mesh, error estimates for the distance of the computable state to the desired target are derived. The main advantage of this new approach is, that applying sparse factorization techniques, a solver of optimal, i.e., almost linear, complexity is proposed and analyzed. The theoretical results are complemented by numerical examples, including discontinuous and less regular targets. Moreover, this approach can be applied also to optimal control problems subject to non-linear state equations.

翻译：本文考虑以抛物型演化方程为约束的分布最优控制问题。控制量在各向异性Sobolev空间$[H_{0;,0}^{1,1/2}(Q)]^\ast$的能量范数下进行考察，使得偏微分方程的状态方程定义了到$H^{1,1/2}_{0;0,}(Q)$的同构映射。由此，可从待极小化的跟踪型泛函中消去控制量，进而导出用于确定状态量的最优性系统。由于所出现的算子诱导出$H_{0;0,}^{1,1/2}(Q)$中的等价范数，我们将通过改进的Hilbert变换，用各向异性Sobolev范数的可计算实现来替代该算子。由此，我们能够将代价或正则化参数$\varrho>0$与状态量和期望目标之间的距离相关联，该距离仅取决于目标的正则性。对于协调的时空有限元离散格式，该性质可延续至离散情形，从而得到正则化参数$\varrho$相对于空间有限元网格尺寸$h_x$的最优选择$\varrho = h_x^2$。利用时空张量积网格，推导了可计算状态量与期望目标之间距离的误差估计。该新方法的主要优势在于，通过应用稀疏分解技术，提出并分析了具有最优（即近乎线性）复杂度的求解器。理论结果通过数值算例（包括间断性及低正则性目标）得到验证。此外，该方法还可应用于受非线性状态方程约束的最优控制问题。