Mathematical Control and Related Fields (MCRF)

Optimal $L^2$-control problem in coefficients for a linear elliptic equation. I. Existence result

Pages: 73 - 96, Volume 5, Issue 1, March 2015      doi:10.3934/mcrf.2015.5.73

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Thierry Horsin - Conservatoire National des Arts et M├ętiers, 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)

Abstract: In this paper we study an optimal control problem (OCP) associated to a linear elliptic equation on a bounded domain $\Omega$. The matrix-valued 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 non-uniqueness 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 well-possed 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 so-called variational and non-variational 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.
Mathematics Subject Classification:  Primary: 49J20, 35J57; Secondary: 49J45, 35J75.

Received: March 2013;      Revised: May 2014;      Available Online: January 2015.