This paper presents a numerical method for the optimal control problem governed by the heat diffusion equation inside a composite medium. The contact resistance at the interface of constitute materials allows for jumps of the temperature field. The derivation process of the Karush-Kuhn-Tucher system is given by the formal Lagrange method. Due to the discontinuity of the temperature field, the standard linear finite element method cannot achieve optimal convergence when the uniform mesh is used. Therefore, the unfitted finite element method is applied to discrete the state equation required in the variational discretization approach. Optimal error estimates in the broken H¹-norm and L²-norm for the control, state, and adjoint state are derived. Some numerical examples are provided to confirm the theoretical results.

本文提出了一种基于复合介质内部热扩散方程的最优控制问题的数值方法。在构成材料的界面处的接触电阻允许温度场的跳跃。Karush-Kuhn-Tucher系统的推导过程通过形式化的Lagrange方法给出。由于温度场的不连续性，使用均匀网格时，标准线性有限元方法无法实现最佳收敛。因此，将不适合的有限元方法应用于离散化离散化方法中所需的状态方程。破裂的H ^1-范数和L ²中的最佳误差估计派生用于控件，状态和伴随状态的-norm。提供了一些数值例子来证实理论结果。