Tunneling Estimates and Approximate Controllability for Hypoelliptic Equations
Seiten
2022
American Mathematical Society (Verlag)
978-1-4704-5138-7 (ISBN)
American Mathematical Society (Verlag)
978-1-4704-5138-7 (ISBN)
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.
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.
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.
Camille Laurent, CNRS, Paris, France, and Sorbonne Universite, Paris, France. Matthieu Leautaud, Ecole Polytechnique, Palaiseau, France.
| Erscheinungsdatum | 31.05.2022 |
|---|---|
| Reihe/Serie | Memoirs of the American Mathematical Society |
| Verlagsort | Providence |
| Sprache | englisch |
| Maße | 178 x 254 mm |
| Gewicht | 203 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Analysis |
| ISBN-10 | 1-4704-5138-7 / 1470451387 |
| ISBN-13 | 978-1-4704-5138-7 / 9781470451387 |
| Zustand | Neuware |
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