We study an optimal control problem governed by elliptic PDEs with interface, which the control acts on the interface. Due to the jump of the coefficient across the interface and the control acting on the interface, the regularity of solution of the control problem is limited on the whole domain, but smoother on subdomains. The control function with pointwise inequality constraints is served as the flux jump condition which we called Neumann interface control. We use a simple uniform mesh that is independent of the interface. The standard linear finite element method can not achieve optimal convergence when the uniform mesh is used. Therefore the state and adjoint state equations are discretized by piecewise linear immersed finite element method (IFEM). While the accuracy of the piecewise constant approximation of the optimal control on the interface is improved by a postprocessing step which possesses superconvergence properties; as well as the variational discretization concept for the optimal control is used to improve the error estimates. Optimal error estimates for the control, suboptimal error estimates for state and adjoint state are derived. Numerical examples with and without constraints are provided to illustrate the effectiveness of the proposed scheme and correctness of the theoretical analysis.

翻译：我们研究了一个由界面椭圆型偏微分方程支配的最优控制问题，其中控制作用施加在界面上。由于系数在界面上的跳跃以及控制作用在界面上，控制问题解在整个区域上的正则性受限，但在子区域上更光滑。具有逐点不等式约束的控制函数作为通量跳跃条件，我们称之为Neumann界面控制。我们采用与界面无关的简单均匀网格。当使用均匀网格时，标准线性有限元方法无法达到最优收敛。因此，状态方程和伴随状态方程采用分片线性浸入式有限元方法(IFEM)进行离散化。同时，通过具有超收敛性质的后处理步骤改进了界面上最优控制的分片常数近似精度；此外，还利用了最优控制的变分离散化概念来改进误差估计。我们导出了控制量的最优误差估计以及状态量和伴随状态量的次优误差估计。提供带约束和不带约束的数值算例，以说明所提方案的有效性和理论分析的正确性。