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.
Author(s): Jérôme Le Rousseau, Gilles Lebeau, Luc Robbiano
Series: Progress in Nonlinear Differential Equations and Their Applications, 97
Publisher: Birkhäuser
Year: 2022
Language: English
Pages: 409
City: Basel
Contents
Part 1. Calculus with a Large Parameter, Carleman Estimates Derivation
Chapter 1. Introduction
1.1. Some Aspects of Unique Continuation
1.2. Form of Carleman Estimates and Quantification of Unique Continuation
1.3. Application to Stabilization and Controllability
1.4. Outline
1.5. Missing Subjects
1.6. Acknowledgement
1.7. Some Notation
1.7.1. Open Sets
1.7.2. Euclidean Inner Products and Norms
1.7.3. Differential Operators
1.7.4. Fourier Transformation
1.7.5. Function Norms
1.7.6. Homogeneity and Conic Sets
1.7.7. Miscellaneous
Chapter 2. (Pseudo-)Differential Operators with a Large Parameter
2.1. Introduction
2.2. Classes of Symbols
2.2.1. Homogeneous and Polyhomogeneous Symbols
2.3. Classes of Pseudo-Differential Operators
2.4. Oscillatory Integrals
2.5. Symbol Calculus
2.6. Sobolev Spaces and Operator Bound
2.7. Positivity Inequalities of Gårding Type
2.8. Parametrices
2.9. Action of Change of Variables
2.10. Tangential Operators
2.11. Semi-Classical Operators
2.12. Standard Pseudo-Differential Operators
2.13. Notes
Appendix
2.A. Technical Proofs for Pseudo-Differential Calculus
2.A.1. Symbol Asymptotic Series: Proof of Lemma 2.4
2.A.2. Action on the Schwartz Space: Proof of Proposition 2.10
2.A.3. Proofs of Results on Oscillatory Integrals
2.A.3.1. Definitions of Oscillatory Integrals: Proof of Theorem 2.11
2.A.3.2. Definitions of Oscillatory Integrals: Proof of Theorem 2.16
2.A.4. Proofs of the Results on Symbol Calculus
2.A.5. Proof of Theorem 2.26: Sobolev Bound
2.A.6. Proofs of the Gårding Inequalities
2.A.6.1. Proof of the Local Gårding Inequality of Theorem 2.28
2.A.6.2. Proof of the Microlocal Gårding Inequality of Theorem 2.29
2.A.6.3. Proof of the Gårding Inequalities for Systems
2.A.7. Parametrix Construction and Properties
2.A.8. A Characterization of Ellipticity
Chapter 3. Carleman Estimate for a Second-Order Elliptic Operator
3.1. Setting
3.2. Weight Function and Conjugated Operator
3.2.1. Conjugated Operator
3.2.2. Characteristic Set and Sub-ellipticity Property
3.2.3. Invariance Under Change of Variables
3.3. Local Estimate Away from Boundaries
3.4. Local Estimates at the Boundary
3.4.1. Some Remarks
3.4.2. Proofs in Adapted Local Coordinates
3.5. Patching Estimates
3.6. Global Estimates with Observation Terms
3.6.1. A Global Estimate with an Inner Observation Term
3.6.2. A Global Estimate with a Boundary Observation Term
3.7. Alternative Approach
3.7.1. A Modified Carleman Estimate Derivation Away from Boundaries
3.7.2. A Modified Carleman Estimate Derivation at a Boundary
3.7.3. Alternative Derivation in the Case of Limited Smoothness
3.7.4. Valuable Aspects of the Different Approaches
3.8. Notes
Appendices
3.A. Poisson Bracket and Weight Function
3.A.1. Smoothness of the Characteristic Set
3.A.2. Expression of the Poisson Bracket
3.A.3. Construction of a Weight Function
3.A.4. Local Extension of the Domain Where Sub-ellipticity Holds
3.B. Symbol Positivity
3.B.1. Symbol Positivity Away from a Boundary
3.B.2. Tangential Symbol Positivity Near a Boundary
3.B.3. Proof of Lemma 3.27
3.B.4. Symbol Positivity in the Modified Approach
3.C. An Explicit Computation
Chapter 4. Optimality Aspects of Carleman Estimates
4.1. On the Necessity of the Sub-ellipticity Property
4.1.1. Bracket Nonnegativity
4.1.2. Optimal Strength in the Large Parameter and Bracket Positivity
4.2. Limiting Weights and Limiting Carleman Estimates
4.2.1. Limiting Weights
4.2.2. Convexification
4.2.3. Limiting Carleman Estimates Away from a Boundary
4.2.4. Global Limiting Carleman Estimates
4.3. Carleman Weight Behavior at a Boundary
4.4. Notes
Appendix
4.A. Some Technical Results
4.A.1. A Linear Algebra Lemma
4.A.2. Sub-ellipticity for First-Order Operators with Linear Symbols
4.A.3. A Particular Class of Limiting Weights
Part 2. Applications of Carleman Estimates
Chapter 5. Unique Continuation
5.1. Introduction
5.2. Local and Global Unique Continuation
5.3. Quantification of Unique Continuation
5.3.1. Quantified Unique Continuation Away from a Boundary
5.3.2. Quantified Unique Continuation Up to a Boundary
5.4. Unique Continuation Initiated at the Boundary
5.5. Unique Continuation Without Any Prescribed Boundary Condition
5.6. Notes
Appendix
5.A. A Hardy Inequality
Chapter 6. Stabilization of the Wave Equation with an Inner Damping
6.1. Introduction and Setting
6.2. Preliminaries on the Damped Wave Equation
6.3. Stabilization and Resolvent Estimate
6.4. Remarks and Non-Quantified Stabilization Results
6.4.1. Comparison with Exponential Stability
6.4.2. Zero Eigenvalue
6.4.3. Non-Quantified Stabilization Results
6.5. Resolvent Estimate for the Damped Wave Generator
6.5.1. Estimations Through an Interpolation Inequality
6.5.2. Estimations Through the Derivation of a Global Carleman Estimate
6.6. Alternative Proof Scheme of the Resolvent Estimate
6.7. Notes
Appendices
6.A. The Generator of the Damped-Wave Semigroup
6.B. Well-Posedness of the Damped Wave Equation
6.B.1. Proof of Well-Posedness
6.B.2. Other Formulations of Weak Solutions
6.C. From a Resolvent to a Semigroup Stabilization Estimate
6.D. Proofs of Non-Quantified Stabilization Results
6.D.1. Proof of Proposition 6.12
6.D.2. Proof of Proposition 6.14
6.D.3. Proof of Proposition 6.15
Chapter 7. Controllability of Parabolic Equations
7.1. Introduction and Setting
7.2. Exact Controllability for a Parabolic Equation
7.3. Null-Controllability for Semigroup Operators
7.4. Observability for the Semigroup Parabolic Equation
7.5. A Spectral Inequality
7.5.1. Spectral Inequality Through an Interpolation Inequality
7.5.2. Spectral Inequality Through the Derivation of a Global Carleman Estimate
7.5.3. Sharpness of the Spectral Inequality
7.6. Partial Observability and Partial Control
7.7. Construction of a Control Function for a Parabolic Equation
7.8. Dual Approach for Observability and Control Cost
7.9. Properties of the Reachable Set and Generalizations
7.10. Boundary Null-Controllability for Parabolic Equations
7.11. Notes
Part 3. Background Material: Analysis and Evolution Equations
Chapter 8. A Short Review of Distribution Theory
8.1. Distributions on an Open Set of Rd and on a Manifold
8.1.1. Test Functions
8.1.2. Definition of Distributions and Basic Properties
8.1.2.1. Localization and Support
8.1.2.2. Distributions with Compact Support
8.1.3. Composition by Diffeomorphisms, Distributions on aManifold
8.2. Temperate Distributions on Rd and Fourier Transformation
8.2.1. The Schwartz Space and Temperate Distributions
8.2.2. The Fourier Transformation on S(Rd), S'(Rd), and L2(Rd)
8.3. Distributions on a Product Space
8.3.1. Tensor Products of Functions
8.3.2. Tensor Products of Distributions
8.3.3. Convolution
8.3.4. The Kernel Theorem (Various Forms)
8.4. Notes
Chapter 9. Invariance Under Change of Variables
9.1. A Review of the Actions of Change of Variables
9.1.1. Pullbacks and Push-Forwards
9.1.2. Action of a Change of Variables on a Differential Operator
9.2. Action on Symplectic Structures
9.2.1. The Symplectic Two-Form
9.2.2. Hamiltonian Vector Fields
9.2.3. Poisson Bracket
9.3. Invariance of the Sub-ellipticity Condition
9.3.1. Action of a Change of Variables on the Conjugated Operator
9.3.2. The Sub-ellipticity Condition
9.4. Normal Geodesic Coordinates
Chapter 10. Elliptic Operator with Dirichlet Data and Associated Semigroup
10.1. Resolvent and Spectral Properties of Elliptic Operators
10.1.1. Basic Properties of Second-Order Elliptic Operators
10.1.2. Spectral Properties
10.1.3. A Sobolev Scale and Operator Extensions
10.2. The Parabolic Semigroup
10.2.1. Spectral Representation of the Semigroup
10.2.2. Well-Posedness: An Elementary Proof
10.2.3. Additional Properties of the Parabolic Semigroup
10.2.4. Properties of the Parabolic Kernel
10.3. The Nonhomogeneous Parabolic Cauchy Problem
10.3.1. Properties of the Duhamel Term
10.3.2. Abstract Solutions of the Nonhomogeneous Semigroup Equations
10.3.3. Strong Solutions
10.3.4. Weak Solutions
10.4. Elementary Form of the Maximum Principle
10.5. The Dirichlet Lifting Map
10.6. Parabolic Equation with Dirichlet Boundary Data
Chapter 11. Some Elements of Functional Analysis
11.1. Linear Operators in Banach Spaces
11.2. Continuous and Bounded Operators
11.3. Spectrum of a Linear Operator in a Banach Space
11.4. Adjoint Operator
11.5. Fredholm Operators
11.5.1. Characterization of Bounded Fredholm Operators
11.6. Linear Operators in Hilbert Spaces
Chapter 12. Some Elements of Semigroup Theory
12.1. Strongly Continuous Semigroups
12.1.1. Definition and Basic Properties
12.1.2. The Hille–Yosida Theorem
12.1.3. The Lumer–Phillips Theorem
12.2. Differentiable and Analytic Semigroups
12.3. Mild Solution of the Inhomogeneous Cauchy Problem
12.4. The Case of a Hilbert Space
Bibliography
Index
Index of notation
