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<p>In this work, we prove the pointwise null controllability of a one-dimensional degenerate parabolic equation with drift and a singular potential, alongside suitable weighted boundary conditions. We show necessary and sufficient conditions for approximate and null controllability of our system. We use a spectral decomposition of a suitable operator, defined in the weighted Lebesgue space <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41980_2024_951_Article_IEq1.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^2_{\beta }(0,1)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>L</mi> <mi>β</mi> <mn>2</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, for appropriate <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41980_2024_951_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\beta \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation>, and the moment method by Fattorini and Russell to get the pointwise null controllability. We also show the null controllability of our system by means of a distributed control of the form <i>h</i>(<i>x</i>)<i>f</i>(<i>t</i>).</p>

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Pointwise Controllability for a 1D Degenerate Parabolic Equation with Drift, and a Singular Potential

  • 期刊论文

  • 发表日期：

    2025年2月11日

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  • Leandro Galo-Mendoza,
  • Marcos López-García

摘要

In this work, we prove the pointwise null controllability of a one-dimensional degenerate parabolic equation with drift and a singular potential, alongside suitable weighted boundary conditions. We show necessary and sufficient conditions for approximate and null controllability of our system. We use a spectral decomposition of a suitable operator, defined in the weighted Lebesgue space \(L^2_{\beta }(0,1)\) L β 2 ( 0 , 1 ) , for appropriate \(\beta \) β , and the moment method by Fattorini and Russell to get the pointwise null controllability. We also show the null controllability of our system by means of a distributed control of the form h(x)f(t).