Problems with control in the coefficients of elliptic equations
Let \(D\subset {\mathbb{R}}^ n\) be a bounded domain with sufficiently smooth boundary \(\Gamma\) and \(V\subset {\mathbb{R}}^ m\) be a bounded set. The following problem is considered: \[ \int_{D}| u_ 1(x)-u_ 2(x)|^ 2dx+\alpha \int_{D}| v(x)-v^ 0(x)|^ 2dx\to \min, \] -(\(\partial /\partial x_ i)a_{ij}(x,v)u_{sx_ j}+a(x,v)u_ s=f(x,v)\), \(x\in D\), \(s=1,2\), \(u_ 1|_{\Gamma}=g_ 1\), \(\partial u_ 2/\partial \nu |_{\Gamma}=g_ 2\), \(v(x)=(v_ 1(x),...,v_ m(x))\in V,\) \(x\in D\), \(u_ s\in W^ 1_ 2(D)\), \(s=1,2\). Here \(\nu\) is conormal and \(a_{ij} (i,j=1,...,n)\), \(a,f,g_ 1,g_ 2,v^ 0\equiv(v^ 0_ 1,...,v^ 0_ m)\) are given functions. It is assumed that the equations above are uniformly elliptic for every admissible v and that the functions \(a_{ij} (i,j=1,...,n)\) a,f are continuous with respect to (x,v). Differentiability of the cost functional and necessary conditions for optimality in form of the maximum principle are investigated. The paper contains inaccuracies and incorrect proofs.
scientific article; zbMATH DE number 3853845