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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Hlavní autoři: Laurent, Camille, Léautaud, Matthieu
Médium: E-kniha Kniha
Jazyk:angličtina
Vydáno: Providence, Rhode Island American Mathematical Society 2022
Vydání:1
Edice:Memoirs of the American Mathematical Society
Témata:
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
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Abstract 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 controllability of the hypoelliptic heat equation We also explain how the analyticity assumption can be relaxed, and a boundary 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-Léautaud (2019).
AbstractList 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 controllability of the hypoelliptic heat equation We also explain how the analyticity assumption can be relaxed, and a boundary 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-Léautaud (2019).
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Author Laurent, Camille
Léautaud, Matthieu
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Keywords eigenfunctions
heat equation
control theory
hypoelliptic operators
Stability estimates
wave equation
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Notes Includes bibliographical references (p. 91-95)
March 2022, volume 276, number 1357 (fifth of 7 numbers)
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Snippet This memoir is concerned with quantitative unique continuation estimates for equations involving a “sum of squares” operator The first result is the tunneling...
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SubjectTerms 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
TableOfContents 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
Title Tunneling estimates and approximate controllability for hypoelliptic equations
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