线性椭圆型方程带系数的最优L2-control问题
Optimal L2-control problem in coefficients for a linear elliptic equation
摘要：
本文研究有界区域上线性椭圆型方程{$Ω$ }的最优控制问题。这样的系统的矩阵值系数A是我们在L2中取得的控制，特别是可以包括无界性的某些情况。对于这类方程的边值问题，可能会遇到弱解的非唯一性，即可逼近解以及不能通过矩阵A的有界逼近得到的另一类弱解.根据变分法中的直接法，我们证明了给定的OCP是适定的，因为它至少允许一个解。同时，此类问题的最优解在上述意义下可能具有奇异性。有鉴于此，我们指出了上述问题的两类最优解：所谓的变分解和非变分解，并证明了某些最优解可以通过特殊的最优边界控制问题的解得到。
英文摘要：
In this paper we study an optimal control problem (OCP) associated to a linear elliptic equation {on a bounded domain $Ω$}. The matrix-valued coefficients A of such systems is our control taken in L2 which in particular may comprise som cases of unboundedness. Concerning the boundary value problems associated to the equations of this type, one may face non-uniqueness of weak solutions--- namely, approximable solutions as well as another type of weak solutions that can not be obtained through the bouded-approximation of matrix A. Following the direct method in the calculus of variations, we show that the given OCP is well-posed in the sense that it admits at least one solution. At the same time, optimal solutions to such problem may have a singular character in the above sense. In view of this, we indicate two types of optimal solutions to the above problem: the so-called variational and non-variational solutions, and show that some of that optimal solutions can be attainable by solutions of special optimal boundary control problems.
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