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 on a compact manifold M assuming: (i) the Chow-Rashevski-Hörmander condition ensuring the hypoellipticity of L, and (ii) the analyticity of M and the coefficients of L. k The first result is the tunneling estimate ||ϕ||L2(ω) ≥ Ce−cλ2 for normalized eigenfunctions ϕ of L from a nonempty open set ω ⊂ M, where k is the hypoellipticity index of L and λ the eigenvalue. The main result is a stability estimate for solutions to the hypoelliptic wave equation (∂t2 + L)u = 0: for T > 2 supx∈M(dist(x, ω)) (here, dist is the sub-Riemannian distance), the observation of the solution on (0, T) × ω determines the data. The constant involved in the estimate is CecΛk where Λ is the typical frequency of the data. We then prove the approximate controllability of the hypoelliptic heat equation (∂t + L)v = 1ωf 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 ∂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). © 2022 Camille Laurent and Matthieu Léautaud