Domain Decomposition Methods In Optimal

While domain decomposition methods have a long history dating back well over one hundred years, it is only during the last decade that they havebecome a major tool in numerical analysis of partial differential equations.This monograph considers problems of optimal control for partial differential equations of elliptic and, more importantly, of hyperbolic types on networked domains. The main goal is to describe, develop and analyze iterative space and time domain decompositions of such problems on the infinite dimensional level.

Preface ix

1 Introduction

1

(8)

2 Background Material on Domain Decomposition

2.1 Introduction

9

(1)

2.2 Domain Decomposition for 1-d Problems

10

(7)

2.2.1 Unbounded Domains

10

(4)

2.2.2 Bounded Domains

14

(1)

2.2.3 Semi-discretization

14

(3)

2.3 Domain Decomposition Methods for Elliptic Problems

17

(54)

2.3.1 Review of Basic Methods

17

(10)

2.3.2 Virtual Controls

27

(13)

2.3.3 The Basic Algorithm of P.-L. Lions

40

(6)

2.3.4 An Augmented Lagrangian Formulation

46

(2)

2.3.5 General Elliptic Problems and More General Splittings

48

(6)

2.3.6 An a Posteriori Error Estimate

54

(3)

2.3.7 Interpretation as a Damped Richardson iteration

57

(7)

2.3.8 A Serial One-Dimensional Problem

64

(7)

3 Partial Differential Equations on Graphs

3.1 Introduction

71

(1)

3.2 Partial Differential Operators on Graphs

72

(4)

3.3 Elliptic Problems on Graphs

76

(15)

3.3.2 Domain Decomposition

79

(3)

3.3.3 Convergence

82

(4)

3.3.4 Interpretation as a Richardson Iteration

86

(5)

3.4 Hyperbolic Problems on Graphs

91

(16)

3.4.1 The Model

91

(5)

3.4.2 The Domain Decomposition Procedure

96

(11)

4 Optimal Control of Elliptic Problems

4.1 Introduction

107

(2)

4.2 Distributed Controls

109

(11)

4.2.2 Domain Decomposition

111

(1)

4.2.3 A Complex Helmholtz Problem and its Decomposition .

111

(2)

4.2.4 Convergence

113

(3)

4.2.5 Methods for Elliptic Optimal Control Problems

116

(1)

4.2.6 An A Posteriori Error Estimate

117

(3)

4.3 Boundary Controls

120

(11)

4.3.2 Domain Decomposition

120

(1)

4.3.3 Convergence

121

(4)

4.3.4 An A Posteriori Error Estimate

125

(6)

5 Control of Partial Differential Equations on Graphs

5.1 Introduction

131

(1)

5.2 Elliptic Problems

131

(15)

5.2.1 The Global Optimal Control Problem on a Graph

131

(2)

5.2.2 Domain Decomposition

133

(2)

5.2.3 Distributed Controls

135

(6)

5.2.4 Boundary Controls

141

(5)

5.3 Hyperbolic Problems

146

(13)

5.3.1 The Global Optimal Control Problem on a Graph

146

(2)

5.3.2 The Domain Decomposition Procedure

148

(11)

6 Control of Dissipative Wave Equations

6.1 Introduction

159

(1)

6.2 Optimal Dissipative Boundary Control

160

(13)

6.2.1 Setting the Problem

160

(2)

6.2.2 Existence and Regularity of Solutions

162

(10)

6.2.3 The Global Optimality System

172

(1)

6.3 Time Domain Decomposition

173

(35)

6.3.1 Description of the Algorithm

173

(6)

6.3.2 Convergence of the Iterates

179

(9)

6.3.3 A Posteriori Error Estimates

188

(13)

6.3.4 Extension to General Dissipative Control Systems

201

(7)

6.4 Decomposition of the Spatial Domain

208

(31)

6.4.1 Description of the Algorithm

208

(8)

6.4.2 Convergence of the Iterates

216

(15)

6.4.3 A Posteriori Error Estimates

231

(8)

6.5 Space and Time Domain Decomposition

239

(18)

6.5.1 Sequential Space-Time Domain Decomposition

239

(11)

6.5.2 Sequential Time-Space Domain Decomposition

250

(7)

7 Boundary Control of Maxwell's System

7.1 Introduction

257

(1)

7.2 Optimal Dissipative Boundary Control

258

(8)

7.2.1 Setting the Problem

258

(2)

7.2.2 Existence and Uniqueness of Solution

260

(5)

7.2.3 The Global Optimality System

265

(1)

7.3 Time Domain Decomposition

266

(23)

7.3.1 Description of the Algorithm

266

(4)

7.3.2 Convergence of the Iterates

270

(7)

7.3.3 A Posteriori Error Estimates

277

(12)

7.4 Decomposition of the Spatial Domain

289

(24)

7.4.1 Description of the Algorithm

289

(4)

7.4.2 Convergence of the Iterates

293

(13)

7.4.3 A Posteriori Error Estimates

306

(7)

7.5 Time and Space Domain Decomposition

313

(8)

7.5.1 Sequential Space-Time Domain Decomposition

313

(4)

7.5.2 Sequential Time-Space Domain Decomposition

317

(4)

8 Control of Conservative Wave Equations

8.1 Introduction

321

(1)

8.2 Optimal Boundary Control

322

(5)

8.2.1 Setting the Problem

322

(2)

8.2.2 Existence and Regularity of Solutions

324

(2)

8.2.3 The Global Optimality System

326

(1)

8.3 Time Domain Decomposition

327

(13)

8.3.1 Description of the Algorithm

327

(2)

8.3.2 Convergence of the Iterates

329

(4)

8.3.3 A Posteriori Error Estiniates

333

(4)

8.3.4 Extension to General Conservative Control Systems

337

(3)

8.4 Decomposition of the Spatial Domain

340

(13)

8.4.1 The Local Optimality Systems

340

(2)

8.4.2 The Domain Decomposition Algorithm

342

(3)

8.4.3 Convergence of the Iterates

345

(8)

8.5 The Exact Reachability Problem

353

(22)

8.5.1 The Global Optimality System

353

(2)

8.5.2 The Limit of the Local Optimality Systems

355

(10)

8.5.3 Application to Domain Decomposition

365

(4)

8.5.4 Convergence to the Global Optimality System

369

(6)

9 Domain Decomposition for 2-D Networks

9.1 Elliptic Systems on 2-D Networks

375

(20)

9.1.2 Examples

378

(6)

9.1.3 Existence and Uniqueness of Solutions

384

(2)

9.1.4 Domain Decomposition

386

(2)

9.1.5 Convergence of the Algorithm

388

(7)

9.2 Optimal Control on 2-D Networks

395

(5)

9.2.1 Optimal Final Value Control

395

(2)

9.2.2 Existence and Regularity of Solutions

397

(3)

9.3 Decomposition of the Spatial Domain

400

(35)

9.3.2 The Decomposition Algorithm

403

(6)

9.3.3 Convergence of the Algorithm

409

(26)

Bibliography

435

(6)

Index

441

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Domain Decomposition Methods In Optimal Control of Partial Differential Equations

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Table of Contents

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Domain Decomposition Methods In Optimal Control of Partial Differential Equations

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Table of Contents

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