In this paper we discuss the numerical solution of elliptic distributed optimal control problems with state or control constraints when the control is considered in the energy norm. As in the unconstrained case we can relate the regularization parameter and the finite element mesh size in order to ensure an optimal order of convergence which only depends on the regularity of the given target, also including discontinuous target functions. While in most cases, state or control constraints are discussed for the more common $L^2$ regularization, much less is known in the case of energy regularizations. But in this case, and for both control and state constraints, we can formulate first kind variational inequalities to determine the unknown state, from wich we can compute the control in a post processing step. Related variational inequalities also appear in obstacle problems, and are well established both from a mathematical and a numerical analysis point of view. Numerical results confirm the applicability and accuracy of the proposed approach.

翻译：本文讨论了在能量范数下考虑控制时，具有状态或控制约束的椭圆型分布最优控制问题的数值解法。与无约束情况类似，我们可以建立正则化参数与有限元网格尺寸之间的联系，从而确保最优收敛阶仅依赖于给定目标的正则性，包括不连续目标函数的情况。虽然在大多数情况下，状态或控制约束是针对更常见的$L^2$正则化进行讨论的，但对于能量正则化的情况所知甚少。然而，在这种情况下，对于控制约束和状态约束，我们均可以构造第一类变分不等式来确定未知状态，并由此通过后处理步骤计算出控制量。相关变分不等式也出现在障碍问题中，且从数学和数值分析角度均已得到充分研究。数值结果验证了所提出方法的适用性和精度。