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<p>In this article, we explore the homogenization of an optimal control problem driven by a semi-linear parabolic equation within a two-dimensional oscillating domain, denoted as <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1045_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Omega _{\epsilon }\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Ω</mi> <mi>ϵ</mi> </msub> </math></EquationSource> </InlineEquation>. The state equation and cost function in this scenario involve periodic coefficients, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1045_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\(A^{\epsilon }\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>A</mi> <mi>ϵ</mi> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1045_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(B^{\epsilon }\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>B</mi> <mi>ϵ</mi> </msup> </math></EquationSource> </InlineEquation>, which exhibit significant oscillations. The objective of this study is to analyze the limiting behavior of both the optimal control and the corresponding state as the oscillations become increasingly fine. Furthermore, we aim to identify the optimal control problem that encapsulates the effects of these oscillating coefficients and to establish a corrector result for the state variable.</p>

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Optimal control and homogenization of semi-linear parabolic problem with highly oscillatory coefficients in an oscillating domain

  • 期刊论文

  • 发表日期：

    2025年3月28日

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  • A. K. Nandakumaran,
  • Ritu Raj,
  • Bidhan Chandra Sardar

摘要

In this article, we explore the homogenization of an optimal control problem driven by a semi-linear parabolic equation within a two-dimensional oscillating domain, denoted as \(\Omega _{\epsilon }\) Ω ϵ . The state equation and cost function in this scenario involve periodic coefficients, \(A^{\epsilon }\) A ϵ and \(B^{\epsilon }\) B ϵ , which exhibit significant oscillations. The objective of this study is to analyze the limiting behavior of both the optimal control and the corresponding state as the oscillations become increasingly fine. Furthermore, we aim to identify the optimal control problem that encapsulates the effects of these oscillating coefficients and to establish a corrector result for the state variable.