Carleman Estimates And Applications To Uniqueness And Control Theory

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Carleman Estimates And Applications To Uniqueness And Control Theory

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Author : Feruccio Colombini
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06

Carleman Estimates And Applications To Uniqueness And Control Theory written by Feruccio Colombini and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.

The articles in this volume reflect a subsequent development after a scientific meeting entitled Carleman Estimates and Control Theory, held in Cartona in September 1999. The 14 research-level articles, written by experts, focus on new results on Carleman estimates and their applications to uniqueness and controlla bility of partial differential equations and systems. The main topics are unique continuation for elliptic PDEs and systems, con trol theory and inverse problems. New results on strong uniqueness for second or higher order operators are explored in detail in several papers. In the area of control theory. the reader will find applications of Carleman estimates to stabiliza tion, observability and exact control for the wave and the SchrOdinger equations. A final paper presents a challenging list of open problems on the topic of control lability of linear and sernilinear heat equations. The papers contain exhaustive and essentially self-contained proofs directly ac cessible to mathematicians, physicists, and graduate students with an elementary background in PDEs. Contributors are L. Aloui, M. Bellassoued, N. Burq, F. Colombini, B. Dehman, C. Grammatico, M. Khenissi, H. Koch, P. Le Borgne, N. Lerner, T. Nishitani. T. Okaji, K.D. Phung, R. Regbaoui, X. Saint Raymond, D. Tataru, and E. Zuazua.

Carleman Estimates And Applications To Uniqueness And Control Theory

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Author : Feruccio Colombini
language : en
Publisher: Springer Science & Business Media
Release Date : 2001-06-21

Carleman Estimates And Applications To Uniqueness And Control Theory written by Feruccio Colombini and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001-06-21 with Mathematics categories.

The articles in this volume reflect a subsequent development after a scientific meeting entitled Carleman Estimates and Control Theory, held in Cartona in September 1999. The 14 research-level articles, written by experts, focus on new results on Carleman estimates and their applications to uniqueness and controlla- bility of partial differential equations and systems. The main topics are unique continuation for elliptic PDEs and systems, con- trol theory and inverse problems. New results on strong uniqueness for second or higher order operators are explored in detail in several papers. In the area of control theory. the reader will find applications of Carleman estimates to stabiliza- tion, observability and exact control for the wave and the SchrOdinger equations. A final paper presents a challenging list of open problems on the topic of control- lability of linear and sernilinear heat equations. The papers contain exhaustive and essentially self-contained proofs directly ac- cessible to mathematicians, physicists, and graduate students with an elementary background in PDEs. Contributors are L. Aloui, M. Bellassoued, N. Burq, F. Colombini, B. Dehman, C. Grammatico, M. Khenissi, H. Koch, P. Le Borgne, N. Lerner, T. Nishitani. T. Okaji, K.D. Phung, R. Regbaoui, X. Saint Raymond, D. Tataru, and E. Zuazua.

Elliptic Carleman Estimates And Applications To Stabilization And Controllability Volume I

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Author : Jérôme Le Rousseau
language : en
Publisher: Springer Nature
Release Date : 2022-03-28

Elliptic Carleman Estimates And Applications To Stabilization And Controllability Volume I written by Jérôme Le Rousseau and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-03-28 with Mathematics categories.

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.

Elliptic Carleman Estimates And Applications To Stabilization And Controllability Volume Ii

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Author : Jérôme Le Rousseau
language : en
Publisher: Springer Nature
Release Date : 2022-04-22

Elliptic Carleman Estimates And Applications To Stabilization And Controllability Volume Ii written by Jérôme Le Rousseau and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-04-22 with Mathematics categories.

This monograph explores applications of Carleman estimates in the study of stabilization and controllability properties of partial differential equations, including quantified unique continuation, logarithmic stabilization of the wave equation, and null-controllability of the heat equation. Where the first volume derived these estimates in regular open sets in Euclidean space and Dirichlet boundary conditions, here they are extended to Riemannian manifolds and more general boundary conditions. The book begins with the study of Lopatinskii-Sapiro boundary conditions for the Laplace-Beltrami operator, followed by derivation of Carleman estimates for this operator on Riemannian manifolds. Applications of Carleman estimates are explored next: quantified unique continuation issues, a proof of the logarithmic stabilization of the boundary-damped wave equation, and a spectral inequality with general boundary conditions to derive the null-controllability result for the heat equation. Two additional chapters consider some more advanced results on Carleman estimates. The final part of the book is devoted to exposition of some necessary background material: elements of differential and Riemannian geometry, and Sobolev spaces and Laplace problems on Riemannian manifolds.

Carleman Inequalities

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Author : Nicolas Lerner
language : en
Publisher: Springer
Release Date : 2019-05-18

Carleman Inequalities written by Nicolas Lerner and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-05-18 with Mathematics categories.

Over the past 25 years, Carleman estimates have become an essential tool in several areas related to partial differential equations such as control theory, inverse problems, or fluid mechanics. This book provides a detailed exposition of the basic techniques of Carleman Inequalities, driven by applications to various questions of unique continuation. Beginning with an elementary introduction to the topic, including examples accessible to readers without prior knowledge of advanced mathematics, the book's first five chapters contain a thorough exposition of the most classical results, such as Calderón's and Hörmander's theorems. Later chapters explore a selection of results of the last four decades around the themes of continuation for elliptic equations, with the Jerison-Kenig estimates for strong unique continuation, counterexamples to Cauchy uniqueness of Cohen and Alinhac & Baouendi, operators with partially analytic coefficients with intermediate results betweenHolmgren's and Hörmander's uniqueness theorems, Wolff's modification of Carleman's method, conditional pseudo-convexity, and more. With examples and special cases motivating the general theory, as well as appendices on mathematical background, this monograph provides an accessible, self-contained basic reference on the subject, including a selection of the developments of the past thirty years in unique continuation.

Carleman Estimates And Applications To Inverse Problems For Hyperbolic Systems

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Author : Mourad Bellassoued
language : en
Publisher: Springer
Release Date : 2017-11-23

Carleman Estimates And Applications To Inverse Problems For Hyperbolic Systems written by Mourad Bellassoued and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-11-23 with Mathematics categories.

This book is a self-contained account of the method based on Carleman estimates for inverse problems of determining spatially varying functions of differential equations of the hyperbolic type by non-overdetermining data of solutions. The formulation is different from that of Dirichlet-to-Neumann maps and can often prove the global uniqueness and Lipschitz stability even with a single measurement. These types of inverse problems include coefficient inverse problems of determining physical parameters in inhomogeneous media that appear in many applications related to electromagnetism, elasticity, and related phenomena. Although the methodology was created in 1981 by Bukhgeim and Klibanov, its comprehensive development has been accomplished only recently. In spite of the wide applicability of the method, there are few monographs focusing on combined accounts of Carleman estimates and applications to inverse problems. The aim in this book is to fill that gap. The basic tool is Carleman estimates, the theory of which has been established within a very general framework, so that the method using Carleman estimates for inverse problems is misunderstood as being very difficult. The main purpose of the book is to provide an accessible approach to the methodology. To accomplish that goal, the authors include a direct derivation of Carleman estimates, the derivation being based essentially on elementary calculus working flexibly for various equations. Because the inverse problem depends heavily on respective equations, too general and abstract an approach may not be balanced. Thus a direct and concrete means was chosen not only because it is friendly to readers but also is much more relevant. By practical necessity, there is surely a wide range of inverse problems and the method delineated here can solve them. The intention is for readers to learn that method and then apply it to solving new inverse problems.

Control And Inverse Problems

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Author : Kaïs Ammari
language : en
Publisher: Springer Nature
Release Date : 2023-09-26

Control And Inverse Problems written by Kaïs Ammari and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-09-26 with Mathematics categories.

This volume presents a timely overview of control theory and inverse problems, and highlights recent advances in these active research areas. The chapters are based on talks given at the spring school "Control & Inverse Problems” held in Monastir, Tunisia in May 2022. In addition to providing a snapshot of these two areas, chapters also highlight breakthroughs on more specific topics, such as: Controllability of dynamical systems Information transfer in multiplier equations Nonparametric instrumental regression Control of chained systems The damped wave equation Control and Inverse Problems will be a valuable resource for both established researchers as well as more junior members of the community.

Carleman Estimates For Second Order Partial Differential Operators And Applications

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Author : Xiaoyu Fu
language : en
Publisher: Springer Nature
Release Date : 2019-10-31

Carleman Estimates For Second Order Partial Differential Operators And Applications written by Xiaoyu Fu and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-10-31 with Mathematics categories.

This book provides a brief, self-contained introduction to Carleman estimates for three typical second order partial differential equations, namely elliptic, parabolic, and hyperbolic equations, and their typical applications in control, unique continuation, and inverse problems. There are three particularly important and novel features of the book. First, only some basic calculus is needed in order to obtain the main results presented, though some elementary knowledge of functional analysis and partial differential equations will be helpful in understanding them. Second, all Carleman estimates in the book are derived from a fundamental identity for a second order partial differential operator; the only difference is the choice of weight functions. Third, only rather weak smoothness and/or integrability conditions are needed for the coefficients appearing in the equations. Carleman Estimates for Second Order Partial Differential Operators and Applications will be of interest to all researchers in the field.

Global Carleman Estimates For Degenerate Parabolic Operators With Applications

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Author : P. Cannarsa
language : en
Publisher: American Mathematical Soc.
Release Date : 2016-01-25

Global Carleman Estimates For Degenerate Parabolic Operators With Applications written by P. Cannarsa and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-01-25 with Mathematics categories.

Degenerate parabolic operators have received increasing attention in recent years because they are associated with both important theoretical analysis, such as stochastic diffusion processes, and interesting applications to engineering, physics, biology, and economics. This manuscript has been conceived to introduce the reader to global Carleman estimates for a class of parabolic operators which may degenerate at the boundary of the space domain, in the normal direction to the boundary. Such a kind of degeneracy is relevant to study the invariance of a domain with respect to a given stochastic diffusion flow, and appears naturally in climatology models.

Unique Continuation Properties For Partial Differential Equations

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Author : Sergio Vessella
language : en
Publisher: Springer Nature
Release Date : 2025-04-29

Unique Continuation Properties For Partial Differential Equations written by Sergio Vessella and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-04-29 with Mathematics categories.

This book provides a comprehensive and self-contained introduction to the study of the Cauchy problem and unique continuation properties for partial differential equations. Aimed at graduate and advanced undergraduate students, it bridges foundational concepts such as Lebesgue measure theory, functional analysis, and partial differential equations with advanced topics like stability estimates in inverse problems and quantitative unique continuation. By presenting detailed proofs and illustrative examples, the text equips readers with a deeper understanding of these fundamental topics and their applications in mathematical analysis. Designed to serve as both a learning resource and a reference, this book is particularly suited for those pursuing research in mathematical physics, inverse problems, or applied analysis.