On approximation of an optimal boundary control problem for linear elliptic equation with unbounded coefficients

Peter I. Kogut

doi:10.3934/dcds.2014.34.2105

Article Contents

2014, Volume 34, Issue 5: 2105-2133. Doi: 10.3934/dcds.2014.34.2105

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On approximation of an optimal boundary control problem for linear elliptic equation with unbounded coefficients

Peter I. Kogut^1,

1.
Basque Center for Applied Mathematics (BCAM), Bizkaia Technology Park, Building 500, E-48160 Derio, Basque Country

Received: February 28, 2013

Revised: July 31, 2013

Published: April 2014

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Abstract

We study an optimal boundary control problem (OCP) associated to a linear elliptic equation $-\mathrm{div}\,\left(\nabla y+A(x)\nabla y\right)=f$. The characteristic feature of this equation is the fact that the matrix $A(x)=[a_{ij}(x)]_{i,j=1,\dots,N}$ is skew-symmetric, $a_{ij}(x)=-a_{ji}(x)$, measurable, and 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--- namely, they have approximable solutions as well as another type of weak solutions that can not be obtained through an approximation of matrix $A$, the corresponding OCP is well-possed and admits a unique solution. At the same time, an optimal solution to such problem can inherit a singular character of the original matrix $A$. We indicate two types of optimal solutions to the above problem: the so-called variational and non-variational solutions, and show that each of that optimal solutions can be attainable by solutions of special optimal boundary control problems.

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Mathematics Subject Classification: Primary: 49J20, 35J57; Secondary: 49J45, 35J75.

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