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Tunneling Estimates and Approximate Controllability for Hypoelliptic Equations

Camille Laurent
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      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-H¨ormander 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) × ? 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.
      Format: Paperback / softback CONTRIBUTORS: Camille Laurent EAN: 9781470451387 COUNTRY: United States PAGES: WEIGHT: 203 g HEIGHT: 254 cm
      PUBLISHED BY: American Mathematical Society DATE PUBLISHED: 2022-06-30 CITY: GENRE: MATHEMATICS / Differential Equations / General WIDTH: 178 cm SPINE:

      Book Themes:

      Differential calculus and equations

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      Camille Laurent, CNRS, Paris, France, and Sorbonne Universite, Paris, France.Matthieu Leautaud, Ecole Polytechnique, Palaiseau, France.
      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-H¨ormander 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) × ? 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.
      Format: Paperback / softback CONTRIBUTORS: Camille Laurent EAN: 9781470451387 COUNTRY: United States PAGES: WEIGHT: 203 g HEIGHT: 254 cm
      PUBLISHED BY: American Mathematical Society DATE PUBLISHED: 2022-06-30 CITY: GENRE: MATHEMATICS / Differential Equations / General WIDTH: 178 cm SPINE:

      Book Themes:

      Differential calculus and equations

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      Be the first to write a review
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      Camille Laurent, CNRS, Paris, France, and Sorbonne Universite, Paris, France.Matthieu Leautaud, Ecole Polytechnique, Palaiseau, France.

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