Title: Tunneling estimates and approximate controllability for hypoelliptic equations,
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-Hörmander 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.