The paper studies the asymptotic behaviour of optimal control for a boundary value problem in a perforated domain with linear Robin boundary conditions, as the period of the structure is <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2025_7965_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">${\varepsilon }\rightarrow 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and parameters of the problem take on the so-called critical values. It is assumed that the cost functional depends on the quadratic form <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2025_7965_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="95" /> </InlineMediaObject> <EquationSource Format="TEX">$(B\nabla {u_{\varepsilon }},\nabla {u_{\varepsilon }})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>B</mi> <mi mathvariant="normal">∇</mi> <msub> <mi>u</mi> <mi>ε</mi> </msub> <mo>,</mo> <mi mathvariant="normal">∇</mi> <msub> <mi>u</mi> <mi>ε</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where B is an arbitrary symmetric positive definite matrix with constant coefficients <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2025_7965_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$b_{i,j}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>b</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2025_7965_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="103" /> </InlineMediaObject> <EquationSource Format="TEX">$i,j=1,\dots , n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation>. Our goal is to show how the presence of the matrix B affects the limit of the control and the cost functional when the period of the structure tends to zero.

ON HOMOGENIZATION FOR CRITICAL PARAMETERS OF AN OPTIMAL PROBLEM IN A PERFORATED DOMAIN WHEN THE COST FUNCTIONAL CONTAINS THE ENERGY INTEGRAL OF A GENERAL FORM

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发表日期：

2025年8月1日

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M. N. Zubova,
T. A. Shaposhnikova

摘要

The paper studies the asymptotic behaviour of optimal control for a boundary value problem in a perforated domain with linear Robin boundary conditions, as the period of the structure is ${\varepsilon }\rightarrow 0$ $ε \to 0$ and parameters of the problem take on the so-called critical values. It is assumed that the cost functional depends on the quadratic form $(B\nabla {u_{\varepsilon }},\nabla {u_{\varepsilon }})$ $(B \nabla u_{ε}, \nabla u_{ε})$ , where B is an arbitrary symmetric positive definite matrix with constant coefficients $b_{i,j}$ $b_{i, j}$ , $i,j=1,\dots , n$ $i, j = 1, \dots, n$ . Our goal is to show how the presence of the matrix B affects the limit of the control and the cost functional when the period of the structure tends to zero.