<p>In this paper, we delve into the intricacies of boundary stabilization for the linearized KP-II equation within the constraints of a bounded domain, a phenomenon known as “critical length.” Our primary aim is to design a feedback law that ensures the existence and exponential stabilization of solutions in the energy space, without length restrictions on the domain <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10883_2025_9733_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="134" /> </InlineMediaObject> <EquationSource Format="TEX">$ \Omega = (0, L) \times (0, L) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>L</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>L</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10883_2025_9733_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">$ L > 0 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. Furthermore, we examine the interaction between the drift term <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10883_2025_9733_Article_IEq3.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$ u_x $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>u</mi> <mi>x</mi> </msub> </math></EquationSource> </InlineEquation> under these constraints.</p>

Boundary Exponential Stabilization for the Linear KP-II Equation without Critical Size Restrictions

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2025年5月5日

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F.A Gallego,
J.R Muñoz

摘要

In this paper, we delve into the intricacies of boundary stabilization for the linearized KP-II equation within the constraints of a bounded domain, a phenomenon known as “critical length.” Our primary aim is to design a feedback law that ensures the existence and exponential stabilization of solutions in the energy space, without length restrictions on the domain $ \Omega = (0, L) \times (0, L) $ $Ω = (0, L) \times (0, L)$ , $ L > 0 $ $L > 0$ . Furthermore, we examine the interaction between the drift term $ u_x $ $u_{x}$ under these constraints.