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Title
Elliptic Carleman estimates and applications to stabilization and controllability. Volume I, Dirichlet boundary conditions on Euclidean space / Jérôme Le Rousseau, Gilles Lebeau, Luc Robbiano.
ISBN
9783030886745 (electronic bk.)
3030886743 (electronic bk.)
9783030886738 (print)
3030886735 (print)
Published
Cham, Switzerland : Birkhäuser, 2022.
Language
English
Description
1 online resource (viii, 411 pages) : illustrations (some color).
Item Number
10.1007/978-3-030-88674-5 doi
Call Number
QA331
Dewey Decimal Classification
515/.9
Summary
This monograph explores applications of Carleman estimates in the study of stabilization and controllability properties of partial differential equations, including the stabilization property of the damped wave equation and the null-controllability of the heat equation. All analysis is performed in the case of open sets in the Euclidean space; a second volume will extend this treatment to Riemannian manifolds. The first three chapters illustrate the derivation of Carleman estimates using pseudo-differential calculus with a large parameter. Continuation issues are then addressed, followed by a proof of the logarithmic stabilization of the damped wave equation by means of two alternative proofs of the resolvent estimate for the generator of a damped wave semigroup. The authors then discuss null-controllability of the heat equation, its equivalence with observability, and how the spectral inequality allows one to either construct a control function or prove the observability inequality. The final part of the book is devoted to the exposition of some necessary background material: the theory of distributions, invariance under change of variables, elliptic operators with Dirichlet data and associated semigroup, and some elements from functional analysis and semigroup theory.
Bibliography, etc. Note
Includes bilbiographical references and indexes.
Access Note
Access limited to authorized users.
Source of Description
Online resource; title from PDF title page (SpringerLink, viewed April 5, 2022).
Series
Progress in nonlinear differential equations and their applications. PNLDE Subseries in Control ; v. 97.
Introduction
Part 1: Calculus with a Large Parameter, Carleman Estimates Derivation
(Pseudo-)differential Operators with a Large Parameter
Carleman Estimate for a Second-Order Elliptic Operator
Optimality Aspects of Carleman Estimates
Part 2: Applications of Carleman Estimates
Unique Continuation
Stabilization of the Wave Equation with an Inner Damping
Controllability of Parabolic Equations
Part 3: Background Material: Analysis and Evolution Equations
A Short Review of Distribution Theory
Invariance under Change of Variables
Elliptic Operator with Dirichlet Data and Associated Semigroup
Some Elements of Functional Analysis
Some Elements of Semigroup Theory
Bibliography
Subject Index
Index of Notation.
