2017/03/31 by Camille Laurent, Laurent, Camille, Matthieu Léautaud +1 · 3 citations
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Optimization and Control (math.OC) #Spectral Theory (math.SP) #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1703.10797
openalex publication_date 2017/03/31 · openalex created_date 2022/09/01 · openalex updated_date 2026/07/28
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‖L2(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 (∂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 developed by the authors in arxiv:1506.04254.