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Non null-controllability properties of the Grushin-like heat equation on 2D-manifolds

Roman Vanlaere

TL;DR

The paper investigates null-controllability for the heat equation with a sub-Laplacian Δ on 2D almost-Riemannian manifolds featuring an interior singularity, under the Grushin-like local structure. It develops a comprehensive spectral framework for the generalized Grushin operator, employing Fourier decomposition, Agmon-type estimates, and complex-analytic techniques to derive obstructions to observability on tensorized Euclidean domains. Through a local reduction to a Euclidean problem in a tubular neighborhood, the authors transfer the non-controllability results to the manifold setting, proving minimal-time lower bounds or failure of null-controllability depending on γ and the control geometry. The results provide a geometric interpretation of degenerate parabolic control in Grushin-type settings and highlight the delicate interplay between singularity proximity, frequency concentration, and control location. Potential extensions include Grushin-type operators beyond warped products, and higher-dimensional or manifold-wide variations with related open problems.

Abstract

We study the internal non null-controllability properties of the heat equation on 2-dimensional almost-Riemannian manifolds with an interior singularity, and under the assumption that the closure of the control zone does not contain the whole singularity. We show that if locally, around the singularity, the sub-Riemannian metric can be written in a Grushin form, or equivalently the sub-Laplacian writes as a generalized Grushin operator, then, achieving null-controllability requires at least a minimal amount of time. As locally the manifold looks like a rectangular domain, we consequently focus ourselves on the non null-controllability properties of the generalized Grushin-like heat equation on various Euclidean domains.

Non null-controllability properties of the Grushin-like heat equation on 2D-manifolds

TL;DR

The paper investigates null-controllability for the heat equation with a sub-Laplacian Δ on 2D almost-Riemannian manifolds featuring an interior singularity, under the Grushin-like local structure. It develops a comprehensive spectral framework for the generalized Grushin operator, employing Fourier decomposition, Agmon-type estimates, and complex-analytic techniques to derive obstructions to observability on tensorized Euclidean domains. Through a local reduction to a Euclidean problem in a tubular neighborhood, the authors transfer the non-controllability results to the manifold setting, proving minimal-time lower bounds or failure of null-controllability depending on γ and the control geometry. The results provide a geometric interpretation of degenerate parabolic control in Grushin-type settings and highlight the delicate interplay between singularity proximity, frequency concentration, and control location. Potential extensions include Grushin-type operators beyond warped products, and higher-dimensional or manifold-wide variations with related open problems.

Abstract

We study the internal non null-controllability properties of the heat equation on 2-dimensional almost-Riemannian manifolds with an interior singularity, and under the assumption that the closure of the control zone does not contain the whole singularity. We show that if locally, around the singularity, the sub-Riemannian metric can be written in a Grushin form, or equivalently the sub-Laplacian writes as a generalized Grushin operator, then, achieving null-controllability requires at least a minimal amount of time. As locally the manifold looks like a rectangular domain, we consequently focus ourselves on the non null-controllability properties of the generalized Grushin-like heat equation on various Euclidean domains.

Paper Structure

This paper contains 22 sections, 17 theorems, 164 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The sub-Riemannian setting
  3. The control problem and main result
  4. State of the art
  5. Structure of the paper
  6. Sketch of proofs
  7. Spectral analysis
  8. Preliminaries on the classical Grushin operator
  9. Dissipation speed
  10. Exponential decay of the eigenfunctions
  11. Dissipation speed for the Fourier components of the generalized operator
  12. Real spectral analysis
  13. Complex spectral analysis
  14. The Grushin equation on tensorized domains
  15. Proof of Theorem \ref{['th gamma >= 1 euclidean bounded']}: control on vertical strips
  16. ...and 7 more sections

Key Result

Theorem 1.5

Assume H0 and H0', and that H2-H3 holds in each of the settings of H1. Depending on the value of $\gamma \geq 1$, we have the following statements for system control grushin system M.

Figures (3)

  • Figure 1: The singularity is in red, the control zone in green, and the tubular neighborhood in blue.
  • Figure 2: The yellow pacman shape is $U$. The disk in red is $D\left(0,e^{-(1-\varepsilon)\min(d_{\operatorname{agm}}(-a),d_{\operatorname{agm}}(b))}\right)$, and it contains the dotted disk $D\left(0,e^{-q'(0)(1+\varepsilon)T} \right)$. When $T$ is too small, there exists $z_0 \in D(0,e^{-q'(0)(1+\varepsilon)T}) \setminus \overline{U}$.
  • Figure 3: The reduction process under \ref{['H1']}. The singularity is in red, the control zones in olive, and the tubular neighborhood $\mathcal{U}$ in light blue. The blue paths correspond to $\partial \mathcal{U} \setminus \partial \mathbb{M}$. The dashed part of $\mathcal{U}$ correspond to where the control is supported after applying Proposition \ref{['proposition internal M to U']}. The arrows represent the passage into coordinates representation, with in olive on the tensorized domains the support of the control for the problem of internal controllability, which is the image by $\Phi$ of the dashed parts.

Theorems & Definitions (43)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 1.4: Null-Controllability
  • Theorem 1.5
  • Theorem 2.1: Control in the complement of a rectangle
  • Theorem 2.2: Control on vertical strips
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • ...and 33 more