Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume I

Dirichlet Boundary Conditions on Euclidean Space
 
Edition number: 1st ed. 2022
Publisher: Birkhäuser
Date of Publication:
Number of Volumes: 1 pieces, Book
 
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Product details:

ISBN13:9783030886769
ISBN10:303088676X
Binding:Paperback
No. of pages:411 pages
Size:254x178 mm
Weight:795 g
Language:English
Illustrations: 8 Illustrations, black & white; 20 Illustrations, color; 18 Tables, color
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Short description:

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.

Long description:
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.
Table of Contents:
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.