Tunneling estimates and approximate controllability for hypoelliptic equations

This memoir is concerned with quantitative unique continuation estimates for equations involving a “sum of squares” operator The first result is the tunneling estimate The main result is a stability estimate for solutions to the hypoelliptic wave equation We then prove the approximate controllabilit...

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Bibliographic Details
Main Authors: Laurent, Camille, Léautaud, Matthieu
Format: eBook Book
Language:English
Published: Providence, Rhode Island American Mathematical Society 2022
Edition:1
Series:Memoirs of the American Mathematical Society
Subjects:
Differential equations, Hypoelliptic
Partial differential equations > Close-to-elliptic equations and systems > Hypoelliptic equations. msc
Partial differential equations > Hyperbolic equations and systems > Wave equation. msc
Partial differential equations > Parabolic equations and systems > Heat equation. msc
Partial differential equations > Qualitative properties of solutions > Continuation and prolongation of solutions. msc
Partial differential equations > Spectral theory and eigenvalue problems > Asymptotic distribution of eigenvalues and eigenfunctions. msc
Systems theory; control > Controllability, observability, and system structure > Controllability. msc
Systems theory; control > Controllability, observability, and system structure > Observability. msc
eigenfunctions
heat equation
control theory
hypoelliptic operators
Stability estimates
wave equation
ISBN:1470451387, 9781470451387
ISSN:0065-9266, 1947-6221
Online Access:Get full text
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Table of Contents:
  • Introduction and main results -- The quantitative Holmgren-John theorem of \cite{LL:15} -- The hypoelliptic wave equation, proof of Theorem 1.15 -- The hypoelliptic heat equation -- A partially analytic example: Grushin type operators -- On the optimality: Proof of Proposition 1.14 -- Subelliptic estimates -- Sub-Riemannian norm of normal vectors
  • 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