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<p>In this work, the exponential stability of the nonlocal fractional heat equation is studied. The fractional Laplacian is defined via a singular integral. Using the spectral properties of the fractional Laplacian and a state decomposition, a feedback control is constructed by considering the first <i>N</i> modes and an observer defined via a bounded operator. Different configurations are examined, including interior controller with interior observation, and interior controller with exterior observation. Using the recent result about the simplicity of the eigenvalues (Fall et al. in Calc Var Partial Differ Equ 62(8):233, 2023), some of our stabilization results are valid for <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="245_2025_10266_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">\(s\in (0,1)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, in particular for <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="245_2025_10266_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="83" /> </InlineMediaObject> <EquationSource Format="TEX">\(s\in (0,1/2)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> in which case the fractional heat equation is not null controllable.</p>

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Observer-Based Exponential Stability of the Fractional Heat Equation

  • 期刊论文

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  • 发表日期：

    2025年5月10日

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  • Hugo Parada

摘要

In this work, the exponential stability of the nonlocal fractional heat equation is studied. The fractional Laplacian is defined via a singular integral. Using the spectral properties of the fractional Laplacian and a state decomposition, a feedback control is constructed by considering the first N modes and an observer defined via a bounded operator. Different configurations are examined, including interior controller with interior observation, and interior controller with exterior observation. Using the recent result about the simplicity of the eigenvalues (Fall et al. in Calc Var Partial Differ Equ 62(8):233, 2023), some of our stabilization results are valid for \(s\in (0,1)\) s ∈ ( 0 , 1 ) , in particular for \(s\in (0,1/2)\) s ∈ ( 0 , 1 / 2 ) in which case the fractional heat equation is not null controllable.