Control Theory for Partial Differential Equations: Continuous and Approximation Theories

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ISBN: 9780521434089 | 0521434084
Cover: Hardcover
Copyright: 2/13/2000

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This is the first volume of a comprehensive and up-to-date two-volume treatment of quadratic optimal control theory for partial differential equations over a finite or infinite time horizon, and related differential (integral) and algebraic Riccati equations. Both continuous theory and numerical approximation theory are included. The authors use an abstract space, operator theoretic approach, which is based on semigroups methods, and which is unifying across a few basic classes of evolution. The various abstract frameworks are motivated by, and ultimately directed to, partial differential equations with boundary/point control. Volume 1 includes the abstract parabolic theory (continuous theory and numerical approximation theory) for the finite and infinite cases and corresponding PDE illustrations as well as various abstract hyperbolic settings in the finite case. It presents numerous new results. These volumes will appeal to graduate students and researchers in pure and applied mathematics and theoretical engineering with an interest in optimal control problems.

Preface

page xv

Some Auxiliary Results on Abstract Equations

645

(28)

Mathematical Setting and Standing Assumptions

645

(3)

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

648

(3)

A Lifting Regularity Property When eAt Is a Group

651

(2)

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

653

(7)

Generation and Abstract Trace Regularity under Unbounded Perturbation

660

(3)

Regularity of a Class of Abstract Damped Systems

663

(4)

Illustrations of Theorem 7.6.2.2 to Boundary Damped Wave Equations

667

(6)

Notes on Chapter 7

671

(1)

References and Bibliography

671

(2)

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

673

(92)

Mathematical Setting and Formulation of the Problem

675

(4)

Statement of Main Results

679

(8)

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

687

(27)

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

714

(19)

Proof of Theorem 8.2.2.1: The More Regular Case

733

(3)

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

736

(9)

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

745

(10)

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

755

(10)

Notes on Chapter 8

761

(2)

References and Bibliography

763

(2)

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

765

(154)

Mathematical Setting and Formulation of the Problem

765

(7)

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

772

(4)

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

776

(39)

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

815

(4)

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

819

(6)

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)

825

(14)

Optimal Control Problem with Bounded Control Operator and Unbounded Observation Operator

839

(3)

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

842

(19)

Proof of Regularity Results Needed in Section 9.8

861

(23)

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

884

(17)

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

901

(7)

Proof of (9.9.1.16) in Lemma 9.9.1.1

908

(2)

Proof of (9.9.3.14) in Lemma 9.9.3.1

910

(9)

Notes on Chapter 9

913

(3)

References and Bibliography

916

(3)

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

919

Mathematical Setting and Problem Statement

920

(6)

Statement of the Main Results

926

(2)

Proof of Theorems 10.2.1 and 10.2.2

928

(8)

Proof of Theorem 10.2.3

936

(6)

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

942

(30)

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

972

(17)

Application: Kirchoff Equation with One Boundary Control. Regularity Theory

989

(30)

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

1019

(23)

Application: Schrodinger Equations with Dirichlet Boundary Control. Regularity Theory

1042

Notes on Chapter 10

1059

(6)

Glossary of Selected Symbols for Chapter 10

1065

(1)

References and Bibliography

1065

Contents of Volume I

Background

(10)

Some Function Spaces Used in Chapter 1

(1)

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

(3)

The Extrapolation Space [D(A*)]'

(1)

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

(4)

References and Bibliography

(2)

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

(110)

Mathematical Setting and Formulation of the Problem

(2)

Statement of Main Results

(7)

Orientation

(2)

Proof of Theorem 1.2.1.1 with GLT Closed

(52)

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

(22)

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

(2)

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

(4)

Extension to Unbounded Operators R and G

103

(9)

Proof of Lemma 1.5.1.1(iii)

112

(9)

Notes on Chapter 1

113

(5)

Glossary of Symbols for Chapter 1

118

(1)

References and Bibliography

119

(2)

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

121

(57)

Mathematical Setting and Formulation of the Problem

122

(3)

Statement of Main Results

125

(4)

Proof of Theorem 2.2.1

129

(26)

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)

155

(5)

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

160

(7)

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

167

(1)

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

168

(10)

Notes on Chapter 2

170

(5)

Glossary of Symbols for Chapter 2

175

(1)

References and Bibliography

176

(2)

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

178

(253)

Examples of Partial Differential Equation Problems Satisfying Chapters 1 and 2

179

(1)

Heat Equation with Dirichlet Boundary Control: Riccati Theory

180

(7)

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

187

(7)

Heat Equation with Neumann Boundary Control

194

(10)

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

204

(4)

Kelvin-Voight Platelike Equation with Point Control with Free BC

208

(3)

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

211

(3)

Another Platelike Equation with Point Control and Clamped BC

214

(2)

The Strongly Damped Wave Equation with Point Control and Dirichlet BC

216

(2)

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

218

(3)

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

221

(3)

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

224

(13)

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

237

(11)

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

248

(13)

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

261

(8)

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

269

(9)

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

278

(4)

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

282

(3)

Damped Elastic Operators

285

(11)

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

296

(15)

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

311

(13)

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

324

(22)

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

346

(17)

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

363

(7)

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

370

(32)

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

402

(29)

Notes on Chapter 3

413

(12)

References and Bibliography

425

(6)

Numerical Approximations of Algebraic Riccati Equations

431

(80)

Introduction: Continuous and Discrete Optimal Control Problems

431

(1)

Background Material

431

(15)

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

446

(5)

Perturbation Results

451

(20)

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

471

(13)

Optimal Rates of Convergence

484

(4)

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

488

(7)

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

495

(16)

Notes on Chapter 4

504

(5)

Glossary of Symbols for Chapter 4

509

(1)

References and Bibliography

509

(2)

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

511

(45)

Introductory Approximation Results

511

(10)

Heat Equation with Dirichlet Boundary Control

521

(10)

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

531

(6)

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

537

(7)

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

544

(5)

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

549

(7)

Notes on Chapter 5

554

(1)

Glossary of Symbols for Chapter 5, Section 5.1

554

(1)

References and Bibliography

554

(2)

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

556

Part I: General Case

557

(1)

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

557

(5)

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

562

(8)

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

570

(11)

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

581

(2)

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

583

(2)

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

585

(1)

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

586

(3)

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

589

(1)

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

589

(4)

A Transition Property for w* for γ > γc

593

(2)

A Transition Property for r* for γ > γc

595

(1)

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

596

(2)

Definition of P and Its Properties

598

(2)

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

600

(3)

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

603

(1)

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

604

(2)

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

606

(1)

Proof of Theorem 6.1.3.2

607

(1)

Part II: The Case Where eAT is Stable:

608

(1)

Motivation, Statement of Main Results

608

(4)

Minimization of J over u for w Fixed

612

(4)

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

616

(2)

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

618

(2)

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

620

(2)

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

622

(4)

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

626

(1)

The Riccati Operator, P, for γ > γc

627

(3)

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

630

Notes on Chapter 6

639

(3)

References and Bibliography

642

Index

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Control Theory for Partial Differential Equations: Continuous and Approximation Theories

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