Tunneling estimates and approximate controllability for hypoelliptic equations
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Bibliographic Information
Tunneling estimates and approximate controllability for hypoelliptic equations
（Memoirs of the American Mathematical Society, no. 1357）
American Mathematical Society, c2022
Available at 3 libraries
Note
"March 2022, volume 276, number 1357 (fifth of 7 numbers)"
Includes bibliographical references (p. 91-95)
Description and Table of Contents
Description
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-Hormander condition ensuring the hypoellipticity of L,and(ii) the analyticity of M and the coefficients of L. The first result is the tunneling estimate ?L2(?) ? Ce?c?k 2 for normalized eigenfunctions ? of L from a nonempty open set ? ?M,wherek is the hypoellipticity index of L and ? the eigenvalue. The main result is a stability estimate for solutions to the hypoelliptic wave equation (?2 t + L)u =0:forT>2supx?M(dist(x,?)) (here, dist is the subRiemannian distance), the observation of the solution on (0,T) x ? determines the data. The constant involved in the estimate is Cec?k where?isthetypical frequency of the data. Wethen prove the approximate controllability of the hypoelliptic heat equation (?t +L)v = 1?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 a nalyticity assumption can be relaxed, and a boundary ?Mcan be added in some situations.
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