Boundary null controllability of a class of 2-D degenerate parabolic PDEs

Víctor Hernández-Santamaría; Subrata Majumdar; Luz de Teresa

doi:10.3934/dcds.2025084

Article Contents

2025, Volume 45, Issue 12: 5107-5153. Doi: 10.3934/dcds.2025084

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Boundary null controllability of a class of 2-D degenerate parabolic PDEs

Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, Ciudad Universitaria, 04510 Coyoacán, Ciudad de México, México

^*Corresponding author: Luz de Teresa
^*Corresponding author: Luz de Teresa

Received: January 2025

Revised: May 2025

Early access: June 10, 2025

Published online: June 10, 2025

This work has received support from UNAM-DGAPA-PAPIIT grant IN117525 (Mexico). V. Hernández-Santamaría is supported by the program "Estancias Posdoctorales por México para la Formación y Consolidación de las y los Investigadores por México" of CONAHCYT (Mexico). He also received support from Project CBF2023-2024-116 of CONAHCYT and by UNAM-DGAPA-PAPIIT grants IA100324 and IN102925 (Mexico). Subrata Majumdar is supported by the UNAM Postdoctoral Program (POSDOC).

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Abstract

This article deals with the boundary null controllability of some degenerate parabolic equations posed on a square domain, presenting the first study of boundary controllability for such equations in multidimensional settings. The proof combines two classical techniques: the method of moments and the Lebeau-Robbiano strategy. A key novelty of this work lies in the analysis of boundary control localized on a subset of the boundary where degeneracy occurs. Furthermore, we establish a Kalman rank condition as a full characterization of boundary controllability for coupled degenerate systems. The results are extended to $ N $-dimensional domains, and some other potential extensions, along with open problems, are discussed to motivate further research in this area.

Keywords:

Mathematics Subject Classification: Primary: 93B05, 93B07, 93C20, 93B60; Secondary: 35K65, 30E05.

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Figure 1. The domain $ \Omega $ for equation 1.1, with the operator degenerating along the dashed lines $ \Gamma_1 $ and $ \Gamma_2 $. The red region, denoted by $ \omega $, represents the control set, which is active at the boundary where the system degenerates

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Figure 2. Example of the domain $\Omega$ for equation (5.5) with $N = 3$. The operator degenerates only on the faces $\Gamma_1$, $\Gamma_4$, and $\Gamma_6$. The red region, denoted by $\omega$, represents the control set, which is active at the degenerate boundary

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Figure 3. The domain $\Omega$ for equation 6.1, with the operator degenerating along the dashed lines $\Gamma_1$ and $\Gamma_2$. The red region, denoted by $\omega$, represents the control set, which is active at the non degenerate boundary

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Figure 4. The domain $\Omega = (-1, 0)\times (0, 1)$ for equation 6.2, with the operator degenerating along the dashed lines $\Gamma_1$ and $\Gamma_2$. The red region, denoted by $\omega$, represents the control set, which is active at the non degenerate boundary

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Figure 5. The domain $\Omega$ for equation 6.5, with the operator degenerating along the dashed lines. The red region, denoted by $\omega$, represents the control set, which is active at the level of the point $x_0$

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Figure 6. Example of the domain $\Omega$ for equation (6.10) for $N = 3$. The operator degenerates only on the dashed face of the cylinder. The red region, denoted by $\omega$, represents the control set, which is active at the degenerate boundary

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