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Observability properties of the singular Grushin equation

Roman Vanlaere

TL;DR

This work characterizes observability for the heat equation with Grushin-type operators perturbed by an inverse-square potential, revealing how the singularity and geometry of the observation set control minimal observability times. The authors integrate Fourier decomposition with sharp Carleman estimates to derive upper bounds, and employ Agmon-type spectral decay to obtain matching lower bounds, obtaining exact minimal times in several configurations. A central finding is the dimension- and singularity-strength dependence of observability, including a concrete Agmon-distance expression that governs lower bounds in the generalized setting. The results illuminate the interplay between sub-Riemannian geometry, spectral properties, and controllability, with implications for almost-Riemannian manifolds and potential extensions beyond the classical Grushin framework.

Abstract

We study the observability properties of the Grushin equation with an inverse square potential, whose singularity occurs at the boundary of two-dimensional rectangular domains or in the interior of the domain in higher dimensions. In some specific configurations of the observation set, we obtain the exact minimal time of observability. The analysis we present relies on recent Carleman estimates obtained by K. Beauchard, J. Dardé, and S. Ervedoza. As a byproduct of these results, we observe, for the heat equation associated to the Laplace-Beltrami operator on almost-Riemannian manifolds, a dependence of the minimal time of observability on the dimension of the singularity.

Observability properties of the singular Grushin equation

TL;DR

This work characterizes observability for the heat equation with Grushin-type operators perturbed by an inverse-square potential, revealing how the singularity and geometry of the observation set control minimal observability times. The authors integrate Fourier decomposition with sharp Carleman estimates to derive upper bounds, and employ Agmon-type spectral decay to obtain matching lower bounds, obtaining exact minimal times in several configurations. A central finding is the dimension- and singularity-strength dependence of observability, including a concrete Agmon-distance expression that governs lower bounds in the generalized setting. The results illuminate the interplay between sub-Riemannian geometry, spectral properties, and controllability, with implications for almost-Riemannian manifolds and potential extensions beyond the classical Grushin framework.

Abstract

We study the observability properties of the Grushin equation with an inverse square potential, whose singularity occurs at the boundary of two-dimensional rectangular domains or in the interior of the domain in higher dimensions. In some specific configurations of the observation set, we obtain the exact minimal time of observability. The analysis we present relies on recent Carleman estimates obtained by K. Beauchard, J. Dardé, and S. Ervedoza. As a byproduct of these results, we observe, for the heat equation associated to the Laplace-Beltrami operator on almost-Riemannian manifolds, a dependence of the minimal time of observability on the dimension of the singularity.
Paper Structure (29 sections, 22 theorems, 249 equations, 3 figures)

This paper contains 29 sections, 22 theorems, 249 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Setting and main results for the classical operator
  3. Setting and main results for the generalized operator
  4. Comments on the results and the literature
  5. Structure of the paper
  6. Well-posedness and Fourier decomposition
  7. Well-posedness
  8. Fourier decomposition
  9. Sketch of proofs
  10. Upper bound on the minimal time of null-controllability: Carleman estimates
  11. Lower Bound on the minimal time of null-controllability: Agmon estimates
  12. Boundary observability via Carleman estimates
  13. Heat kernel for the classical operator on the half-line
  14. A global Carleman estimate for the classical operator
  15. Boundary observability
  16. ...and 14 more sections

Key Result

Theorem 1.2

Consider system stm : adjoint system classical with $\gamma = 1$ and $\nu > 0$, and assume either assumption: d = 1 or assumption: d > 3. Set $\omega = \omega_x \times \Omega_y \subset \Omega$ to be an open set such that $0_{\mathbb{R}^{d_x}} \notin \overline{\omega}_x \subset \Omega_x$. Then, the m If moreover $\omega_x$ satisfies that there exists a smooth domain $\mathcal{O} \subset\subset \Ome

Figures (3)

  • Figure 1: The geometric assumption on the observation set in Theorem \ref{['thm: main theorem control classical grushin rectangle vertical']}: the observation set is in green, whose boundary contains the boundary in red of a subdomain $\mathcal{O}$ containing $0$.
  • Figure 2: Exhaustive graph of the cutoffs $\chi_1$ and $\chi_2$.
  • Figure 3: Exhaustive graphs of the functions $\eta_i$.

Theorems & Definitions (38)

  • Definition 1.1: Observability
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Proposition 2.2
  • Proposition 4.1
  • Proposition 4.2
  • proof
  • Proposition 4.3: Boundary Carleman estimate
  • ...and 28 more