Tunneling estimates and approximate controllability for hypoelliptic equations

TL;DR

This paper establishes quantitative estimates for hypoelliptic operators on compact manifolds, including tunneling, controllability, and stability results, with implications for wave and heat equations under analyticity and geometric conditions.

Contribution

It introduces new tunneling and controllability estimates for hypoelliptic equations on manifolds, extending previous methods to operators satisfying the Hörmander condition.

Findings

Eigenfunction tunneling estimate with exponential decay

Stability estimate for hypoelliptic wave equation solutions

Approximate controllability of hypoelliptic heat equation with exponential or polynomial cost

Abstract

This article is concerned with quantitative unique continuation estimates for equations involving a "sum of squares" operator $\mathcal{L}$ on a compact manifold $\mathcal{M}$ assuming: $(i)$ the Chow-Rashevski-H\"ormander condition ensuring the hypoellipticity of $\mathcal{L}$, and $(ii)$ the analyticity of $\mathcal{M}$ and the coefficients of $\mathcal{L}$. The first result is the tunneling estimate $\|\varphi\|_{L^2(\omega)} \geq Ce^{- \lambda^{\frac{k}{2}}}$ for normalized eigenfunctions $\varphi$ of $\mathcal{L}$ from a nonempty open set $\omega\subset \mathcal{M}$, where $k$ is the hypoellipticity index of $\mathcal{L}$ and $\lambda$ the eigenvalue. The main result is a stability estimate for solutions to the hypoelliptic wave equation $(\partial_t^2+\mathcal{L})u=0$: for $T>2 \sup_{x \in \mathcal{M}}(dist(x,\omega))$ (here, $dist$ is the sub-Riemannian distance), the observation of the solution on $(0,T)\times \omega$ determines the data. The constant involved in the estimate is $Ce^{c\Lambda^k}$ where $\Lambda$ is the typical frequency of the data. We then prove the approximate controllability of the hypoelliptic heat equation $(\partial_t+\mathcal{L})v=1_\omega f$ in any time, with appropriate (exponential) cost, depending on $k$. In case $k=2$ (Grushin, Heisenberg...), we further show approximate controllability to trajectories with polynomial cost in large time. We also explain how the analyticity assumption can be relaxed, and a boundary $\partial \mathcal{M}$ can be added in some situations. Most results turn out to be optimal on a family of Grushin-type operators. The main proof relies on the general strategy developed by the authors in arxiv:1506.04254.