The composites of materials with high contrasting properties is an interesting topic to study as it has applications. In this article, we wish to study problems in high oscillating domains, where the oscillatory part is made of two materials with high contrasting conductivities (or diffusivity). Thus the low contrast material acts as near insulation in-between the conducting materials. In the first part, we study the homogenization problem of an elliptic equation. The main discussion in this article is the study of optimal control problems based on the unfolding method. The interesting result is the difference in the limit behavior of the optimal control problem, which crucially depends on the action of the control, whether it is on the conductivity part or insulating part. In both cases, we derive the two-scale limit controls problems which are quite similar as far as analysis is concerned. But, if the controls are acting on the conductive region, a complete scale separation is available, whereas a complete separation is not visible in the insulating case due to the intrinsic nature of the problem. In this case, to obtain the limit optimal control problem in the macro scale, two cross-sectional cell problems are introduced. We do obtain the homogenized equation for the state, but the two-scale cost functional remains as it is.

具有高对比度特性的材料复合材料由于其应用而成为一个有趣的研究主题。在本文中，我们希望研究高振荡域中的问题，其中振荡部分是由两种具有高对比度电导率（或扩散率）的材料制成的。因此，低对比度材料在导电材料之间充当近乎绝缘的作用。在第一部分中，我们研究了椭圆方程的均化问题。本文的主要讨论是基于展开方法的最优控制问题的研究。有趣的结果是，最佳控制问题的极限行为有所不同，这主要取决于控制的作用，无论是导电部分还是绝缘部分。在这两种情况下 我们得出了两尺度极限控制问题，就分析而言，它们非常相似。但是，如果控件作用在导电区域上，则可以实现完全的水垢分离，而由于问题的内在本质，在绝缘情况下看不到完全的水垢分离。在这种情况下，为了获得宏观尺度上的极限最优控制问题，引入了两个横截面单元问题。我们确实获得了状态的均化方程，但是两级成本函数保持原样。为了在宏观尺度上获得极限最优控制问题，引入了两个横截面单元问题。我们确实获得了状态的均化方程，但是两级成本函数保持原样。为了在宏观尺度上获得极限最优控制问题，引入了两个横截面单元问题。我们确实获得了状态的均化方程，但是两级成本函数保持原样。