In this paper, we investigate optimal control problems governed by the parabolic interface equation, in which the control acts on the interface. The solution to this problem exhibits low global regularity due to the jump of the coefficient across the interface and the control acting on the interface. Consequently, the traditional finite element method fails to achieve optimal convergence rates when using a uniform mesh. To discretize the problem, we use fully discrete approximations based on the stable generalized finite element method for spatial discretization and the backward Euler scheme for temporal discretization, as well as variational discretization for the control variable. We prove a priori error estimates for the control, state, and adjoint state. Numerical examples are provided to support the theoretical findings.

翻译：本文研究由抛物界面方程控制的最优控制问题，其中控制作用在界面上。由于系数在界面上的跳跃以及控制作用在界面上，该问题的解具有较低的全局正则性。因此，当使用均匀网格时，传统的有限元方法无法达到最优收敛率。为了离散化该问题，我们采用基于稳定广义有限元方法进行空间离散、向后欧拉格式进行时间离散的完全离散近似，并对控制变量采用变分离散化。我们证明了控制、状态和伴随状态的先验误差估计。数值算例验证了理论结果。