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Optimal L 1-Control in Coefficients for Dirichlet Elliptic Problems: W-Optimal Solutions

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Abstract

In this paper, 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 coefficients may degenerate and, therefore, the problems may exhibit the so-called Lavrentieff phenomenon and non-uniqueness of weak solutions. We consider the solvability of this problem in the class of W-variational solutions. Using a concept of variational convergence of constrained minimization problems in variable spaces, we prove the existence of W-solutions to the optimal control problem and provide the way for their approximation. We emphasize that control problems of this type are important in material and topology optimization as well as in damage or life-cycle optimization.

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Authors and Affiliations

  1. Department of Differential Equations, Dnipropetrovsk National University, Kozakova str., 18/14, 49050, Dnipropetrovsk, Ukraine

    Peter I. Kogut

  2. Institüt für Angewandte Mathematik, Lehrstuhl II Universität Erlangen-Nürnberg, Martensstr. 3, 91058, Erlangen, Germany

    Guenter Leugering

Authors
  1. Peter I. Kogut
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  2. Guenter Leugering
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Correspondence to Guenter Leugering.

Additional information

Communicated by H.J. Pesch.

The authors gratefully acknowledge the support of the Cluster of Excellence EAM “Engineering of advanced materials”.

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Kogut, P.I., Leugering, G. Optimal L 1-Control in Coefficients for Dirichlet Elliptic Problems: W-Optimal Solutions. J Optim Theory Appl 150, 205–232 (2011). https://doi.org/10.1007/s10957-011-9840-4

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  • DOI: https://doi.org/10.1007/s10957-011-9840-4

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