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Abstract <p> The paper studies the asymptotic behavior of the optimal control and the optimal state which is the solution to a boundary value problem defined in a planar domain perforated by small sets (inclusions or particles) of various shapes but with the same perimeter and uniformly bounded diameter. The difference in the shapes of the inclusions makes the structure of the inhomogeneous domain aperiodic. A Robin-type condition is imposed on the boundary of these particles with a coefficient <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\beta(\varepsilon)\)</EquationSource> </InlineEquation>, growing rapidly as <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\varepsilon\to 0\)</EquationSource> </InlineEquation>. Furthermore, it is assumed that the control is imposed on the boundary of only those inclusions that belong to a certain subset <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\omega\)</EquationSource> </InlineEquation> of the original domain <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\Omega\)</EquationSource> </InlineEquation>. The cost functional consists of the Dirichlet integral and the control cost. At a critical relationship between the problem parameters (the diameter of the base cell, the size of the inclusions, and the coefficient in the boundary condition), we show the emergence of “strange” terms in the limit problem for the state and in the limit cost functional. </p>

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Homogenization of an Optimal Control Problem in an Aperiodically Perforated Plane Domain with Control on the Boundary of Part of the Inclusions: A Critical Case

  • 期刊论文

  • 发表日期：

    2025年12月1日

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  • A. V. Podol’skii,
  • T. A. Shaposhnikova

摘要

Abstract

The paper studies the asymptotic behavior of the optimal control and the optimal state which is the solution to a boundary value problem defined in a planar domain perforated by small sets (inclusions or particles) of various shapes but with the same perimeter and uniformly bounded diameter. The difference in the shapes of the inclusions makes the structure of the inhomogeneous domain aperiodic. A Robin-type condition is imposed on the boundary of these particles with a coefficient \(\beta(\varepsilon)\) , growing rapidly as \(\varepsilon\to 0\) . Furthermore, it is assumed that the control is imposed on the boundary of only those inclusions that belong to a certain subset \(\omega\) of the original domain \(\Omega\) . The cost functional consists of the Dirichlet integral and the control cost. At a critical relationship between the problem parameters (the diameter of the base cell, the size of the inclusions, and the coefficient in the boundary condition), we show the emergence of “strange” terms in the limit problem for the state and in the limit cost functional.