In this paper we study we study a Dirichlet optimal control prob- lem associated with a linear elliptic equation the coefficients of which we take as controls in the class of integrable functions. The characteristic feature of this control object is the fact that the skew-symmetric part of matrix-valued control A(x) belongs to L2-space (rather than Linfinty). In spite of the fact that the equations of this type can exhibit non-uniqueness of weak solutions, the corresponding OCP, under rather general assumptions on the class of admissi- ble controls, is well-posed and admits a nonempty set of solutions [9]. However, the optimal solutions to such problem may have a singular character. We show that some of optimal solutions can be attainable by solutions of special optimal control problems in perforated domains with fictitious boundary controls on the holes.

中文翻译：

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线性椭圆方程系数中的最优 $L^2$ 控制问题。二、解的近似和最优条件

在本文中，我们研究了与线性椭圆方程相关的狄利克雷最优控制问题，我们将其系数作为可积函数类的控制。该控制对象的特征是矩阵值控制 A(x) 的偏对称部分属于 L2 空间（而不是 Linfinty）。尽管这种类型的方程可以表现出弱解的非唯一性，但相应的 OCP，在对可接受控制类别的相当普遍的假设下，是适定的，并允许一组非空的解[9] ]。然而，此类问题的最优解可能具有单一性。