Tunneling estimates and approximate controllability for hypoelliptic equations
Résumé
This article 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. The first result is the tunneling estimate ϕ L 2 (ω) ≥ Ce −λ k 2 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 (∂ 2 t +L)u = 0: for T > 2 sup x∈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 Ce cΛ k 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 developed by the authors in [LL15].
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