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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 821964, 13 pages
http://dx.doi.org/10.1155/2013/821964
Research Article

On an Optimal -Control Problem in Coefficients for Linear Elliptic Variational Inequality

1Department of System Analysis and Control, National Mining University, Karl Marx Avenue 19, Dnipropetrovsk 49005, Ukraine
2The Research Laboratory for Nonlinear Analysis of Differential-Operator Systems of Institute for Applied and System Analysis in National Technical University of Ukraine “Kiev Polytechnical Institute”, Peremogy Avenue 37, building 35, Kiev 03056, Ukraine
3Dipartimento di Ingegneria dell’Informazione, Ingegneria Elettrica e Matematica Applicata, Università degli Studi di Salerno, Via Giovanni Paolo II 132, 84084 Fisciano, Italy

Received 27 January 2013; Accepted 24 June 2013

Academic Editor: Matti Lassas

Copyright © 2013 Olha P. Kupenko and Rosanna Manzo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider optimal control problems for linear degenerate elliptic variational inequalities with homogeneous Dirichlet boundary conditions. We take the matrix-valued coefficients in the main part of the elliptic operator as controls in . Since the eigenvalues of such matrices may vanish and be unbounded in , it leads to the “noncoercivity trouble.” Using the concept of convergence in variable spaces and following the direct method in the calculus of variations, we establish the solvability of the optimal control problem in the class of the so-called -admissible solutions.