Title: Regularization and finite element error estimates for elliptic distributed optimal control problems with energy regularization and state or control constraints Show Chinese title

Title: 具有能量正则化和状态或控制约束的椭圆分布式最优控制问题的正则化和有限元误差估计

Abstract: 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. Show Chinese abstract

Abstract: 在本文中，我们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，我们可以将正则化参数和有限元网格大小相关联，以确保收敛的最优阶数，该阶数仅取决于给定目标的正则性，也包括不连续的目标函数。 虽然在大多数情况下，状态或控制约束通常针对更常见的$L^2$正则化进行讨论，但在能量正则化的情况下，了解的内容要少得多。 但在这种情况下，对于控制和状态约束，我们可以建立第一类变分不等式来确定未知的状态，从其中可以在后处理步骤中计算控制。 相关的变分不等式也出现在障碍问题中，并且从数学和数值分析的角度来看已经得到了很好的确立。 数值结果证实了所提出方法的适用性和准确性。