We consider a space-time finite element method on fully unstructured simplicial meshes for optimal sparse control of semilinear parabolic equations. The objective is a combination of a standard quadratic tracking-type functional including a Tikhonov regularization term and of the $L^1$-norm of the control that accounts for its spatio-temporal sparsity. We use a space-time Petrov-Galerkin finite element discretization for the first-order necessary optimality system of the associated discrete optimal sparse control problem. The discretization is based on a variational formulation that employs piecewise linear finite elements simultaneously in space and time. Finally, the discrete nonlinear optimality system that consists of coupled forward-backward state and adjoint state equations is solved by a semismooth Newton method.

中文翻译：

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抛物方程最优稀疏控制的非结构化时空有限元方法

我们考虑在完全非结构化单纯网格上的时空有限元方法，以实现半线性抛物线方程的最优稀疏控制。目标是标准二次跟踪型函数的组合，包括 Tikhonov 正则化项和控制的 $L^1$-范数，该范数解释了其时空稀疏性。我们对相关离散最优稀疏控制问题的一阶必要最优系统使用时空 Petrov-Galerkin 有限元离散化。离散化基于变分公式，该公式在空间和时间上同时采用分段线性有限元。最后，通过半光滑牛顿法求解由耦合的前向-后向状态方程和伴随状态方程组成的离散非线性最优系统。