Elliptic carleman estimates and applications to stabilization and controllability. Volume I, Dirichlet boundary conditions on Euclidean space [electronic resource]

• 作者： Le Rousseau, Jerome.
• 其他作者：
  • Lebeau, Gilles.
  • Robbiano, Luc.
  • SpringerLink (Online service)
• 其他題名：
  • Dirichlet boundary conditions on Euclidean space
  • Progress in nonlinear differential equations and their applications ;
• 出版： Cham : Springer International Publishing :Imprint: Birkhauser
• 叢書名： Progress in nonlinear differential equations and their applications,v. 97
• 主題： Carleman theorem. , Differential equations, Partial. , Differential Equations. , Systems Theory, Control. , Operator Theory. , Functional Analysis.
• ISBN： 9783030886745 (electronic bk.) 、 9783030886738 (paper)
• FIND@SFXID: CGU
• 資料類型: 電子書
• 內容註: 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.
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• 摘要註: 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.
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