In this paper, we analyze optimal control problems governed by an elliptic partial differential equation, in which the control acts as the Dirichlet data. Box constraints for the controls are imposed and the cost functional involves the state and possibly a sparsity-promoting term, but not a Tikhonov regularization term. Two different discretizations are investigated: the variational approach and a full discrete approach. For the latter, we use continuous piecewise linear elements to discretize the control space and numerical integration of the sparsity-promoting term. It turns out that the best way to discretize the state equation is to use the Carstensen quasi-interpolant of the boundary data, and a new discrete normal derivative of the adjoint state must be introduced to deal with this. Error estimates, optimization procedures and examples are provided.

在本文中，我们分析了由椭圆偏微分方程控制的最优控制问题，其中控制作为狄利克雷数据。强加了控件的框约束，成本函数涉及状态和可能的稀疏促进项，但不涉及 Tikhonov 正则化项。研究了两种不同的离散化：变分方法和完全离散方法。对于后者，我们使用连续分段线性元素来离散化控制空间和稀疏促进项的数值积分。事实证明，离散状态方程的最佳方法是使用边界数据的 Carstensen 拟插值，并且必须引入伴随状态的新离散正态导数来处理这个问题。误差估计，