Tunneling Estimates and Approximate Controllability for Hypoelliptic Equations.

Format:

Book

Author/Creator:

Laurent, Camille.

Contributor:

Léautaud, Matthieu.

Series:

Memoirs of the American Mathematical Society

Memoirs of the American Mathematical Society ; v.276

Language:

English

Subjects (All):

Differential equations, Hypoelliptic.

Physical Description:

1 online resource (108 pages)

Edition:

1st ed.

Place of Publication:

Providence : American Mathematical Society, 2022.

Summary:

"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 , and (ii) the analyticity of M and the coefficients of . The first result is the tunneling estimate for normalized eigenfunctions of from a nonempty open set , where is the hypoellipticity index of and the eigenvalue. The main result is a stability estimate for solutions to the hypoelliptic wave equation 2 t + L = 0 for T 2 sup (here, dist is the sub- Riemannian distance), the observation of the solution on (0, T) determines the data. The constant involved in the estimate is Ceck where is the typical frequency of the data. We then prove the approximate controllability of the hypoelliptic heat equation (t +L) = 1 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 to produce quantitative unique continuation estimates, developed by the authors in Laurent-Leautaud (2019)"-- Provided by publisher.

Contents:

Cover

Title page

Chapter 1. Introduction and main results

1.1. Introduction

1.2. Main results

1.3. Comparison to other works

1.4. Sketch of the proofs and plan of the paper

1.5. Some remarks and further comments

Chapter 2. The quantitative Holmgren-John theorem of [LL19]

2.1. A typical quantitative unique continuation result of [LL19]

2.2. Definitions and tools for propagating the information

2.3. Semiglobal estimates along foliation by hypersurfaces

Chapter 3. The hypoelliptic wave equation, proof of Theorem 1.15

3.1. Step 1: Geometric setting and non-characteristic hypersurfaces

3.2. Step 2: Propagation of smallness

3.3. Step 3: Energy estimates

Chapter 4. The hypoelliptic heat equation

4.1. Approximate controllability with polynomial cost in large time: Proof of Theorem 1.22

4.2. Approximate controllability in Gevrey-type spaces: Proof of Theorem 1.20

4.3. Approximate controllability in natural spaces with exponential cost: Proof of Theorem 1.18

4.4. Technical lemmata used for the heat equation

Chapter 5. A partially analytic example: Grushin type operators

5.1. The geometric context

5.2. A proof of Estimate (1.26)

5.3. An observation term in ² in quantitative unique continuation estimates

Appendix A. On the optimality: Proof of Proposition 1.14

Appendix B. Subelliptic estimates

B.1. ^{ } subelliptic estimates on compact manifolds

B.2. Subelliptic estimates for manifolds with boundaries

Appendix C. Sub-Riemannian norm of normal vectors

Bibliography

Back Cover.

Notes:

Description based on publisher supplied metadata and other sources.

Other Format:

Print version: Laurent, Camille Tunneling Estimates and Approximate Controllability for Hypoelliptic Equations

ISBN:

9781470470234

1470470233

OCLC:

1321799636