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Mathematical Control and Related Fields (MCRF)
 

Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions

Pages: 595 - 628, Volume 6, Issue 4, December 2016      doi:10.3934/mcrf.2016017

 
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Thierry Horsin - heSam Université, Conservatoire National des Arts et Métiers, M2N, Case 2D 5000, 292 rue Saint-Martin, 75003 Paris, France (email)
Peter I. Kogut - Department of Differential Equations, Dnipropetrovsk National University, Gagarin av., 72, 49010 Dnipropetrovsk, Ukraine (email)
Olivier Wilk - heSam Université Conservatoire National des Arts et Métiers, M2N, Case 2D 5000, 292 rue Saint-Martin, 75003 Paris, France (email)

Abstract: In this paper we study we study a Dirichlet optimal control problem 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 $L^2$-space (rather than $L^\infty)$. 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 admissible 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.

Keywords:  Control in coefficients, non-variational solutions, variational convergence, fictitious control.
Mathematics Subject Classification:  Primary: 35Q93, 49J20, 49N05, 65N99; Secondary: 49J30, 35J75, 65N20.

Received: October 2015;      Revised: July 2016;      Available Online: October 2016.

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