# Non null controllability of the Grushin equation in 2D

## TL;DR

This paper investigates the null controllability of a 2D Grushin equation, demonstrating that control is impossible when the control region does not intersect certain horizontal bands, using complex analysis and eigenvalue estimates.

## Contribution

It provides a novel analysis linking null controllability of the Grushin equation to entire function estimates and eigenvalue bounds, with new insights into the spectral properties involved.

## Key findings

- Null controllability fails if control region avoids horizontal bands.
- Eigenvalue estimates for a related Schrödinger operator are established.
- Complex domain analysis of eigenvalues enhances understanding of controllability limits.

## Abstract

We are interested in the exact null controllability of the equation $\partial_t f - \partial_x^2 f - x^2 \partial_y^2f = \mathbf 1_\omega u$, with control $u$ supported on $\omega$. We show that, when $\omega$ does not intersect a horizontal band, the considered equation is never null-controllable. The main idea is to interpret the associated observability inequality as an $L^2$ estimate on entire functions, which Runge's theorem disproves. To that end, we study in particular the first eigenvalue of the operator $-\partial_x^2 + (nx)^2$ with Dirichlet conditions on $(-1,1)$ and we show a quite precise estimation it satisfies, even when $n$ is in some complex domain.