Tunneling estimates and approximate controllability for hypoelliptic equations

This memoir is concerned with quantitative unique continuation estimates for equations involving a “sum of squares” operator L \mathcal {L} on a compact manifold M \mathcal {M} assuming: ( i ) (i) the Chow-Rashevski-Hörmander condition ensuring the hypoellipticity of L \mathcal {L} , and ( i i ) (ii) the analyticity of M \mathcal {M} and the coefficients of L \mathcal {L} . The first result is the tunneling estimate ‖ φ ‖ L 2 ( ω ) ≥ C e − c λ k 2 \|\varphi \|_{L^2(\omega )} \geq Ce^{- c\lambda ^{\frac {k}{2}}} for normalized eigenfunctions φ \varphi of L \mathcal {L} from a nonempty open set ω ⊂ M \omega \subset \mathcal {M} , where k k is the hypoellipticity index of L \mathcal {L} and λ \lambda the eigenvalue. The main result is a stability estimate for solutions to the hypoelliptic wave equation ( ∂ t 2 + L ) u = 0 (\partial _t^2+\mathcal {L})u=0 : for T > 2 sup x ∈ M ( d i s t ( x , ω ) ) T>2 \sup _{x \in \mathcal {M}}(dist(x,\omega )) (here, d i s t dist is the sub-Riemannian distance), the observation of the solution on ( 0 , T ) × ω (0,T)\times \omega determines the data. The constant involved in the estimate is C e c Λ k Ce^{c\Lambda ^k} where Λ \Lambda is the typical frequency of the data. We then prove the approximate controllability of the hypoelliptic heat equation ( ∂ t + L ) v = 1 ω f (\partial _t+\mathcal {L})v=\mathbb {1}_\omega f in any time, with appropriate (exponential) cost, depending on k k . In case k = 2 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 ∂ M \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 to produce quantitative unique continuation estimates, developed by the authors in Laurent-Léautaud (2019).