Well-Posed Linear Systems

List offigur es page ix

Preface xi

List ofnotation xiv

1 Introduction and overview 1

1.1 Introduction 1

1.2 Overview of chapters 2–13 8

2 Basic properties of well-posed linear systems 28

2.1 Motivation 28

2.2 Definitions and basic properties 34

2.3 Basic examples of well-posed linear systems 46

2.4 Time discretization 55

2.5 The growth bound 60

2.6 Shift realizations 67

2.7 The Lax–Phillips scattering model 71

2.8 The Weiss notation 76

2.9 Comments 78

3 Strongly continuous semigroups 85

3.1 Norm continuous semigroups 85

3.2 The generator of a C0 semigroup 87

3.3 The spectra of some generators 98

3.4 Which operators are generators? 106

3.5 The dual semigroup 113

3.6 The rigged spaces induced by the generator 122

3.7 Approximations of the semigroup 128

3.8 The nonhomogeneous Cauchy problem 133

3.9 Symbolic calculus and fractional powers 140

3.10 Analytic semigroups and sectorial operators 150

v

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3.11 Spectrum determined growth 164

3.12 The Laplace transform and the frequency domain 169

3.13 Shift semigroups in the frequency domain 177

3.14 Invariant subspaces and spectral projections 180

3.15 Comments 191

4 The generators of a well-posed linear system 194

4.1 Introduction 194

4.2 The control operator 196

4.3 Differential representations of the state 202

4.4 The observation operator 213

4.5 The feedthrough operator 219

4.6 The transfer function and the system node 227

4.7 Operator nodes 238

4.8 Examples of generators 256

4.9 Diagonal and normal systems 260

4.10 Decompositions of systems 266

4.11 Comments 273

5 Compatible and regular systems 276

5.1 Compatible systems 276

5.2 Boundary control systems 284

5.3 Approximations of the identity in the state space 295

5.4 Extended observation operators 302

5.5 Extended observation/feedthrough operators 313

5.6 Regular systems 317

5.7 Examples of regular systems 325

5.8 Comments 329

6 Anti-causal, dual, and inverted systems 332

6.1 Anti-causal systems 332

6.2 The dual system 337

6.3 Flow-inversion 349

6.4 Time-inversion 368

6.5 Time-flow-inversion 378

6.6 Partial flow-inversion 386

6.7 Comments 400

7 Feedback 403

7.1 Static output feedback 403

7.2 Additional feedback connections 413

7.3 State feedback and output injection 422

7.4 The closed-loop generators 425

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7.5 Regularity of the closed-loop system 433

7.6 The dual of the closed-loop system 436

7.7 Examples 436

7.8 Comments 440

8 Stabilization and detection 443

8.1 Stability 443

8.2 Stabilizability and detectability 453

8.3 Coprime fractions and factorizations 465

8.4 Coprime stabilization and detection 473

8.5 Dynamic stabilization 485

8.6 Comments 502

9Realizations 505

9.1 Minimal realizations 505

9.2 Pseudo-similarity of minimal realizations 511

9.3 Realizations based on factorizations of the Hankel operator 517

9.4 Exact controllability and observability 521

9.5 Normalized and balanced realizations 530

9.6 Resolvent tests for controllability and observability 538

9.7 Modal controllability and observability 546

9.8 Spectral minimality 549

9.9 Controllability and observability of transformed systems 551

9.10 Time domain tests and duality 554

9.11 Comments 565

10 Admissibility 569

10.1 Introduction to admissibility 569

10.2 Admissibility and duality 572

10.3 The Paley–Wiener theorem and H∞ 576

10.4 Controllability and observability gramians 583

10.5 Carleson measures 591

10.6 Admissible control and observation operators for diagonal

and normal semigroups 598

10.7 Admissible control and observation operators for

contraction semigroups 602

10.8 Admissibility results based on the Lax–Phillips model 610

10.9 Comments 613

11 Passive and conservative scattering systems 616

11.1 Passive systems 616

11.2 Energy preserving and conservative systems 628

11.3 Semi-lossless and lossless systems 636

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11.4 Isometric and unitary dilations of contraction semigroups 643

11.5 Energy preserving and conservative extensions of

passive systems 655

11.6 The universal model of a contraction semigroup 660

11.7 Conservative realizations 670

11.8 Energy preserving and passive realizations 677

11.9 The Spectrum of a conservative system 691

11.10 Comments 692

12 Discrete time systems 696

12.1 Discrete time systems 696

12.2 The internal linear fractional transform 703

12.3 The Cayley and Laguerre transforms 707

12.4 The reciprocal transform 719

12.5 Comments 728

Appendix 730

A.1 Regulated functions 730

A.2 The positive square root and the polar decomposition 733

A.3 Convolutions 736

A.4 Inversion of block matrices 744

Bibliography 750

Index 767