SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A744-A771, January 2021.
This work presents and analyzes space-time finite element methods on fully unstructured simplicial space-time meshes for the numerical solution of parabolic optimal control problems. Using Babuška's theorem, we show well-posedness of the first-order optimality systems for a typical model problem with linear state equations, but without control constraints. This is done for both continuous and discrete levels. Based on these results, we derive discretization error estimates. Then we consider a semilinear parabolic optimal control problem arising from the Schlögl model. The associated nonlinear optimality system is solved by Newton's method, where a linear system, which is similar to the first-order optimality systems considered for the linear model problems, has to be solved at each Newton step. We present various numerical experiments including results for adaptive space-time finite element discretizations based on residual-type error indicators. In the last two examples, we also consider semilinear parabolic optimal control problems with box constraints imposed on the control.

中文翻译：

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抛物方程最优控制的非结构化时空有限元方法

SIAM科学计算杂志，第43卷，第2期，第A744-A771页，2021年1月。
这项工作提出并分析了完全非结构化简单时空网格上的时空有限元方法，以求解抛物线型最优控制问题的数值解。使用巴布斯卡定理，我们针对具有线性状态方程但没有控制约束的典型模型问题，展示了一阶最优系统的适定性。对于连续和离散级别都可以完成此操作。基于这些结果，我们得出离散化误差估计。然后我们考虑由Schlögl模型引起的半线性抛物线最优控制问题。关联的非线性最优系统是通过牛顿法求解的，其中类似于在线性模型问题中考虑的一阶最优系统的线性系统，必须在每个牛顿步骤中求解。我们提出了各种数值实验，包括基于残差类型误差指标的自适应时空有限元离散化的结果。在最后两个示例中，我们还考虑了对控制施加盒约束的半线性抛物型最优控制问题。