Optimal $L^2$control problem in coefficients for a linear elliptic equation. I. Existence result
Thierry Horsin  Conservatoire National des Arts et Métiers, M2N, Case 2D 5000, 292 rue SaintMartin, 75003 Paris, France (email) Abstract: In this paper we study an optimal control problem (OCP) associated to a linear elliptic equation on a bounded domain $\Omega$. The matrixvalued coefficients $A$ of such systems is our control in $\Omega$ and will be taken in $L^2(\Omega;\mathbb{R}^{N\times N})$ which in particular may comprises the case of unboundedness. Concerning the boundary value problems associated to the equations of this type, one may exhibit nonuniqueness of weak solutions namely, approximable solutions as well as another type of weak solutions that can not be obtained through the $L^\infty$approximation of matrix $A$. Following the direct method in the calculus of variations, we show that the given OCP is wellpossed and 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 socalled variational and nonvariational solutions, and show that some of that optimal solutions can not be attainable through the $L^\infty$approximation of the original problem.
Keywords: Control in coefficients, variational solutions, variational convergence, existence result.
Received: March 2013; Revised: May 2014; Available Online: January 2015. 
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