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
© Cambridge University Press www.cambridge.org
Cambridge University Press
0521825849 - Well-Posed Linear Systems
Olof Staffans
Table of Contents
More information
vi Contents
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
© Cambridge University Press www.cambridge.org
Cambridge University Press
0521825849 - Well-Posed Linear Systems
Olof Staffans
Table of Contents
More information
Contents vii
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
© Cambridge University Press www.cambridge.org
Cambridge University Press
0521825849 - Well-Posed Linear Systems
Olof Staffans
Table of Contents
More information
viii Contents
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
