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

Víctor Hernández-Santamaría, Subrata Majumdar, Luz de Teresa

TL;DR

This work establishes boundary null controllability for a class of 2D degenerate parabolic PDEs on the unit square, where the diffusion degenerates along coordinate directions and control is applied on the degenerate boundary. The authors merge the one-dimensional moment-method controllability with the Lebeau-Robbiano strategy, supported by a Kalman rank condition that fully characterizes controllability for coupled degenerate systems, and provide explicit cost bounds of the form $\|q\|_{L^2(0,T;L^2(\partial\Omega)^m)}\le C e^{C/T}\|u_0\|_{H^{-1}_{\alpha}(\Omega)^n}$. They carry the analysis from 1D to 2D and then extend to $N$ dimensions, using a partial observability inequality and spectral inequalities to glue the arguments. The results yield new insight into the boundary control of multidimensional degenerate parabolic systems and identify open problems for more general degeneracies and geometries. The work advances the theory of boundary controllability for degenerate PDEs with potential extensions to more complex domains and coefficient structures.

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 the degeneracy occurs. Furthermore, we establish the Kalman rank condition as a full characterization of boundary controllability for coupled degenerate systems. The results are extended to $N$-dimensional domains, and potential extensions and open problems are discussed to motivate further research in this area.

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

TL;DR

This work establishes boundary null controllability for a class of 2D degenerate parabolic PDEs on the unit square, where the diffusion degenerates along coordinate directions and control is applied on the degenerate boundary. The authors merge the one-dimensional moment-method controllability with the Lebeau-Robbiano strategy, supported by a Kalman rank condition that fully characterizes controllability for coupled degenerate systems, and provide explicit cost bounds of the form . They carry the analysis from 1D to 2D and then extend to dimensions, using a partial observability inequality and spectral inequalities to glue the arguments. The results yield new insight into the boundary control of multidimensional degenerate parabolic systems and identify open problems for more general degeneracies and geometries. The work advances the theory of boundary controllability for degenerate PDEs with potential extensions to more complex domains and coefficient structures.

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 the degeneracy occurs. Furthermore, we establish the Kalman rank condition as a full characterization of boundary controllability for coupled degenerate systems. The results are extended to -dimensional domains, and potential extensions and open problems are discussed to motivate further research in this area.

Paper Structure

This paper contains 31 sections, 32 theorems, 221 equations, 6 figures.

Table of Contents

  1. Introduction and main results
  2. System under study
  3. State of the art
  4. Statement of the main results
  5. Strategy of the proof of \ref{['main theorem']}
  6. Paper organization
  7. Functional framework and well-posedness
  8. A one-dimensional boundary controllability problem for degenerate systems
  9. Functional setting for the one dimensional case
  10. Weakly degenerate case ($0\leq\alpha_1<1$)
  11. Strongly degenerate case ($1<\alpha_1< 2$)
  12. Spectral analysis of the 1-$d$ degenerate parabolic operator
  13. Well-posedness
  14. Analysis of the coupled degenerate operator
  15. Proof of the main controllability result in 1-$d$
  16. ...and 16 more sections

Key Result

Theorem 1.2

Let $T>0$ and $\alpha_i \in [0,2)\setminus\{1\}$ for $i = 1, 2$. For any $u_0\in H^{-1}_{\alpha}(\Omega)$, there exists a control $q\in L^2(0,T;L^2(\partial \Omega))$ such that system DCP_sc--bd3 satisfies $u(T)=0$.

Figures (6)

  • Figure 1: The domain $\Omega$ for equation \ref{['DCP_sc']}, 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.
  • Figure 2: Example of the domain $\Omega$ for equation \ref{['eq:sys_dim3']} 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
  • Figure 3: The domain $\Omega$ for equation \ref{['DCP2']}, 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.
  • Figure 4: The domain $\Omega=(-1,0)\times (0,1)$ for equation \ref{['DCP3']}, 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.
  • Figure 5: The domain $\Omega$ for equation \ref{['DCP5']}, 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 $x_0$.
  • ...and 1 more figures

Theorems & Definitions (60)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 50 more