Problems with control in the coefficients of elliptic equations
type: Item
instance of: scholarly article
title: Problems with control in the coefficients of elliptic equations
zbMATH Open document ID: 0537.49007
DOI: 10.1007/BF01157410
author: M. D. Madatov
published in: Mathematical Notes
publication date: 1983-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 49K20
Mathematics Subject Classification ID: 35B37
Mathematics Subject Classification ID: 35J55
Mathematics Subject Classification ID: 93C20
zbMATH DE Number: 3853845
zbMATH Keywords: control in the coefficients
zbMATH Keywords: maximum principle
reviewed by: Uldis Raitums
MaRDI profile type: Publication
cites work: On Existence of Optimal Control
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description: scientific article; zbMATH DE number 3853845