Tunneling estimates and approximate controllability for hypoelliptic equations

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.