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 $|\varphi|{L^2(\omega)} \geq Ce^{- \lambda^{\frac{k}{2}}}$fornormalizedeigenfunctions$\varphi$of$$\mathcal{L}$$fromanonemptyopenset$\omega\subset $\mathcal{M}$$,where$k$isthehypoellipticityindexof$$\mathcal{L}$$and$\lambda$ the eigenvalue. The main result is a stability estimate for solutions to the hypoelliptic wave equation $({\mathrm{\partial }}_t^2+\mathcal{L})u=0$: 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 $C{e}^{c{\mathrm{\Lambda }}^k}$ where $\mathrm{\Lambda }$ is the typical frequency of the data. We then prove the approximate controllability of the hypoelliptic heat equation $({\mathrm{\partial }}_t+\mathcal{L})v=1_{\omega }f$ 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 $\mathrm{\partial }\mathcal{M}$ 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.