Preface

Some Auxiliary Results on Abstract Equations

Mathematical Setting and Standing Assumptions

Regularity of L and L* on [0,T]

A Lifting Regularity Property When eAt Is a Group

Extension of Regularity of L and L* on [0, ∞] When eAt Is Uniformly Stable

Generation and Abstract Trace Regularity under Unbounded Perturbation

Regularity of a Class of Abstract Damped Systems

Illustrations of Theorem 7.6.2.2 to Boundary Damped Wave Equations

Notes on Chapter 7

References and Bibliography

Optimal Quadratic Cost Problem Over a Preassigned Finite Time Interval: The Case Where the Input → Solution Map Is Unbounded, but the Input → Observation Map Is Bounded

Mathematical Setting and Formulation of the Problem

Statement of Main Results

The General Case. A First Proof of Theorems 8.2.1.1 and 8.2.1.2 by a Variational Approach: From the Optimal Control Problem to the DRE and the IRE Theorem 8.2.1.3

A Second Direct Proof of Theorem 8.2.1.2: From the Well-Posedness of the IRE to the Control Problem. Dynamic Programming

Proof of Theorem 8.2.2.1: The More Regular Case

Application of Theorems 8.2.1.1, 8.2.1.2, and 8.2.2.1: Neumann Boundary Control and Dirichlet Boundary Observation for Second-Order Hyperbolic Equations

A One-Dimensional Hyperbolic Equation with Dirichlet Control (B Unbounded) and Point Observation (R Unbounded) That Satisfies (h.1) and (h.3) but not (h.2), (H.1),(H.2), and (H.3). Yet, the DRE Is Trivially Satisfied as a Linear Equation

Interior and Boundary Regularity of Mixed Problems for Second-Order Hyperbolic Equations with Neumann-Type BC

Notes on Chapter 8

References and Bibliography

Optimal Quadratic Cost Problem over a Preassigned Finite Time Interval: The Case Where the Input → Solution Map Is Bounded. Differential and Integral Riccati Equations

Mathematical Setting and Formulation of the Problem

Statement of Main Result: Theorems 9.2.1, 9.2.2, and 9.2.3

Proofs of Theorem 9.2.1 and Theorem 9.2.2 (by the Variational Approach and by the Direct Approach). Proof of Theorem 9.2.3

Isomorphism of P(t), 0 ≤ t < T, and Exact Controllability of [A*, R*] on [0, T -- t] When G = 0

Nonsmoothing Observation R: ``Limit Solution'' of the Differential Riccati Equation under the Sole Assumption (A.1) When G = 0

Dual Differential and Intergral Riccati Equations When A is a Group Generator under (A.1) and R ∈ ℒ (Y; Z) and G = 0. (Bounded Control Operator, Unbounded Observation)

Optimal Control Problem with Bounded Control Operator and Unbounded Observation Operator

Application to Hyperbolic Partial Differential Equations with Point Control. Regularity Theory

Proof of Regularity Results Needed in Section 9.8

A Coupled System of a Wave and a Kirchhoff Equation with Point Control, Arising in Noise Reduction. Regularity Theory

A Coupled System of a Wave and a Structurally Damped Euler--Bernoulli Equation with Point Control, Arising in Noise Reduction. Regularity Theory

Proof of (9.9.1.16) in Lemma 9.9.1.1

Proof of (9.9.3.14) in Lemma 9.9.3.1

Notes on Chapter 9

References and Bibliography

Differential Riccati Equations under Slightly Smoothing Observation Operator. Applications to Hyperbolic and Petrowski-Type PDEs. Regularity Theory

Mathematical Setting and Problem Statement

Statement of the Main Results

Proof of Theorems 10.2.1 and 10.2.2

Proof of Theorem 10.2.3

Application: Second-Order Hyperbolic Equations with Dirichlet Boundary Control. Regularity Theory

Application: Nonsymmetric, Nondissipative First-Order Hyperbolic Systems with Boundary Control. Regularity Theory

Application: Kirchoff Equation with One Boundary Control. Regularity Theory

Application: Euler-Bernoulli Equation with One Boundary Control. Regularity Theory

Application: Schrodinger Equations with Dirichlet Boundary Control. Regularity Theory

Notes on Chapter 10

Glossary of Selected Symbols for Chapter 10

References and Bibliography

Contents of Volume I

Background

Some Function Spaces Used in Chapter 1

Regularity of the Variation of Parameter Formula When eAt Is a s.c. Analytic Semigroup

The Extrapolation Space [D(A*)]'

Abstract Setting for Volume I. The Operator LT in (1.1.9), or LsT in (1.4.1.6), of Chapter 1

References and Bibliography

Optimal Quadratic Cost Problem Over a Preassigned Finite Time Interval: Differential Riccati Equation

Mathematical Setting and Formulation of the Problem

Statement of Main Results

Orientation

Proof of Theorem 1.2.1.1 with GLT Closed

First Smoothing Case of the Operator G: The Case (-- A*)β G* G ∈ ℒ(Y), β > 2γ -- 1. Proof of Theorem 1.2.2.1

A Second Smoothing Case of the Operator G: The Case (-- A*)γ G*G ∈ ℒ(Y). Proof of Theorem 1.2.2.2

The Theory of Theorem 1.2.1.1 Is Sharp. Counterexamples When GLT Is Not Closable

Extension to Unbounded Operators R and G

Proof of Lemma 1.5.1.1(iii)

Notes on Chapter 1

Glossary of Symbols for Chapter 1

References and Bibliography

Optimal Quadratic Cost Problem over an Infinite Time Interval: Algebraic Riccati Equation

Mathematical Setting and Formulation of the Problem

Statement of Main Results

Proof of Theorem 2.2.1

Proof of Theorem 2.2.2: Exponential Stability of Φ(t) and Uniqueness of the Solution of the Algebraic Riccati Equation under the Detectability Condition (2.1.13)

Extensions to Unbounded R : R ∈ ℒ(D (Aδ); Z), δ < min {1 -- γ, 1/2}

Bounded Inversion of [I + SV], S, V ≥ 0

The Case &thetas; = 1 in (2.3.7.4) When A is Self-Adjoint and R = I

Notes on Chapter 2

Glossary of Symbols for Chapter 2

References and Bibliography

Illustrations of the Abstract Theory of Chapters 1 and 2 to Partial Differential Equations with Boundary/Point Controls

Examples of Partial Differential Equation Problems Satisfying Chapters 1 and 2

Heat Equation with Dirichlet Boundary Control: Riccati Theory

Heat Equation with Dirichlet Boundary Control: Regularity Theory of the Optimal Pair

Heat Equation with Neumann Boundary Control

A Structurally Damped Platelike Equation with Point Control and Simplified Hinged BC

Kelvin-Voight Platelike Equation with Point Control with Free BC

A Structurally Damped Platelike Equation with Boundary Control in the Simplified Moment BC

Another Platelike Equation with Point Control and Clamped BC

The Strongly Damped Wave Equation with Point Control and Dirichlet BC

A Structurally Damped Kirchhoff Equation with Point Control Acting through δ(. --x0) and Simplified Hinged BC

A Structurally Damped Kirchhoff Equation (Revisited) with Point Control Acting through δ'(. --x0) and Simplified Hinged BC

Thermo-Elastic Plates with Thermal Control and Homogeneous Clamped Mechanical BC

Thermo-Elastic Plates with Mechanical Control in the Bending Moment (Hinged BC) and Homogeneous Neumann Thermal BC

Thermo-Elastic Plates with Mechanical Control as a Shear Force (Free BC)

Structurally Damped Euler-Bernoulli Equations with Damped Free BC and Point Control or Boundary Control

A Linearized Model of Well/Reservoir Coupling for a Monophasic Flow with Boundary Control

Additional Illustrations with Control Operator B and Observation Operator R Both Genuinely Unbounded

Interpolation (Intermediate) Sobolev Spaces and Their Indentification with Domains of Fractional Powers of Elliptic Operators

Damped Elastic Operators

Boundary Operators for Bending Moments and Shear Forces on Two-Dimensional Domains

C0-Semigroup/Analytic Semigroup Generation when A = AM, A Positive Self-Adjoint, M Matrix. Applications to Thermo-Elastic Equations with Hinged Mechanical BC and Dirichlet Thermal BC

Analyticity of the s.c. Semigroups Arisng from Abstract Thermo-Elastic Equations. First Proof

Analyticity of the s.c. Semigroup Arising from Abstract Thermo-Elastic Equations. Second Proof

Analyticity of the s.c. Semigroup Arising from Abstract Thermo-Elastic Equations. Third Proof

Analyticity of the s.c. Semigroup Arising from Problem (3.12.1) (Hinged Mechanical BC/Neumann (Robin) Thermal BC)

Uniform Exponential Energy Decay of Thermo-Elastic Equations with, or without, Rotational Term. Energy Methods

Notes on Chapter 3

References and Bibliography

Numerical Approximations of Algebraic Riccati Equations

Introduction: Continuous and Discrete Optimal Control Problems

Background Material

Convergence Properties of the Operators Lh and Last;h; Lh and L*h

Perturbation Results

Uniform Convergence Ph Πh → P and B*h Ph Πh → B* P

Optimal Rates of Convergence

A Sharp Result on the Exponential Operator-Norm Decay of a Family of Strongly Continuous Semigroups

Finite Element Approximations of Dynamic Compensators of Luenberger's Type for Partially Observed Analytic Systems with Fully Unbounded Control and Observation Operators

Notes on Chapter 4

Glossary of Symbols for Chapter 4

References and Bibliography

Illustrations of the Numerical Theory of Chapter 4 to Parabolic-Like Boundary/Point Control PDE Problems

Introductory Approximation Results

Heat Equation with Dirichlet Boundary Control

Heat Equation with Neumann Boundary Control. Optimal Rates of Convergence with r ≥ 1 and Galerkin Approximation

A Structurally Damped Platelike Equation with Interior Point Control with r ≥ 3

Kelvin--Voight Platelike Equation with Interior Point Control with r ≥ 3

A Structurally Damped Platelike Equation with Boundary Control with r ≥ 3

Notes on Chapter 5

Glossary of Symbols for Chapter 5, Section 5.1

References and Bibliography

Min--Max Game Theory over an Infinite Time Interval and Algebraic Riccati Equations

Part I: General Case

Mathematical Setting; Formulation of the Min-Max Game Problem; Statement of Main Results

Minimization of Jw,T over u ∈ L2(0, T; U) for w Fixed

Minimization of Jw,∞ over u ∈ L2(0, ∞; U) for w Fixed: The Limit Process as T ↑ ∞

Collection of Explicit Formulae for pw,∞, rw,∞, and y0w,∞ in Stable Form

Explicit Expression for the Optimal Cost J0w,∞ (y0 = 0) as a Quadratic Term

Definition of the Critical value γc. Coercivity of Eγ for γ > γc

Maximization of J0w,∞ over w Directly on [0, ∞] for γ > γc. Characterization of Optimal Quantities

Explicit Expression of w*(.; y0) in Terms of the Data via E--1γ for γ > γc

Smoothing Properties of the Operators L, L*, W, W*: The Optimal u*, y*, w* Are Continuous in Time

A Transition Property for w* for γ > γc

A Transition Property for r* for γ > γc

The Semigroup Property for y* and a Transition Property for p* for γ > γc

Definition of P and Its Properties

The Feedback Generator Af and Its Preliminary Properties for γ > γc

The Operator P is a Solution of the Algebraic Riccati Equation, AREγy for γ > γc

The Semigroup Generated by (A -- BB*P) Is Uniformly Stable

The Case 0 < γ < γc: sup J0w,∞(y0) = +∞

Proof of Theorem 6.1.3.2

Part II: The Case Where eAT is Stable:

Motivation, Statement of Main Results

Minimization of J over u for w Fixed

Maximization of J0w(y0) over w: Existence of a Unique Optimal w*

Explicit Expressions of {u*, y*, w*} and P for γ > γc in Terms of the Data via E--1γ

Smoothing Properties of the Operators L, L*, W, W*: The Optimal u*, y*, w* Are Continuous in Time

A Transition Property for w* for γ > &gammxa;c

The Semigroup Property for y* for γ > γc and Its Stability

The Riccati Operator, P, for γ > γc

Optimal Control Problem with Nondefinite Quardratic Cost. The Stable, Analytic Case. A Brief Sketch

Notes on Chapter 6

References and Bibliography

Index