In this paper we propose a new finite element method for solving elliptic optimal control problems with pointwise state constraints, including the distributed controls and the Dirichlet or Neumann boundary controls. The main idea is to use energy space regularizations in the objective functional, while the equivalent representations of the energy space norms, i.e., the $H^{-1}(\Omega)$-norm for the distributed control, the $H^{1/2}(\Gamma)$-norm for the Dirichlet control and the $H^{-1/2}(\Gamma)$-norm for the Neumann control, enable us to transform the optimal control problem into an elliptic variational inequality involving only the state variable. The elliptic variational inequalities are second order for the three cases, and include additional equality constraints for Dirichlet or Neumann boundary control problems. Standard $C^0$ finite elements can be used to solve the resulted variational inequality. We provide preliminary a priori error estimates for the new algorithm for solving distributed control problems. Extensive numerical experiments are carried out to validate the accuracy of the new algorithm.

翻译：本文提出了一种求解具有逐点状态约束的椭圆最优控制问题（包括分布控制、Dirichlet或Neumann边界控制）的新有限元方法。其核心思想是在目标泛函中采用能量空间正则化，同时通过能量空间范数的等价表示——即分布控制的$H^{-1}(\Omega)$范数、Dirichlet控制的$H^{1/2}(\Gamma)$范数和Neumann控制的$H^{-1/2}(\Gamma)$范数——将最优控制问题转化为仅涉及状态变量的椭圆变分不等式。针对三种情形，该椭圆变分不等式均为二阶形式，且在Dirichlet或Neumann边界控制问题中包含额外的等式约束。可采用标准$C^0$有限元求解所得变分不等式。我们给出了新算法求解分布控制问题的初步先验误差估计，并通过大量数值实验验证了新算法的精度。