Well-Posed Linear Systems (Encyclopedia of Mathematics and its Applications)

796 Pages·2005·10.17 MB·English

by  Staffans O., Rota G.C. (ed.), Doran B. (ed.)

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Encyclopedia of Mathematics and Its Applications
Founding Editor G. C. Rota
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Well-Posed Linear Systems
OLOF STAFFANS
DepartmentofMathematics
A˚boAkademiUniversity,Finland
cambridge university press
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
TheEdinburghBuilding,Cambridgecb22ru,UK
Published in the United States of America by Cambridge University Press, New York
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© Cambridge University Press 2005
Thisbookisincopyright.Subjecttostatutoryexceptionandtotheprovisionof
relevantcollectivelicensingagreements,noreproductionofanypartmaytakeplace
withoutthewrittenpermissionofCambridgeUniversityPress.
Firstpublishedinprintformat 2005
isbn-13 978-0-511-08208-5 eBook (NetLibrary)
isbn-10 0-511-08208-8 eBook (NetLibrary)
isbn-13 978-0-521-82584-9 hardback
isbn-10 0-521-82584-9 hardback
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Contents
Listoffigures pageix
Preface xi
Listofnotation xiv
1 Introductionandoverview 1
1.1 Introduction 1
1.2 Overviewofchapters2–13 8
2 Basicpropertiesofwell-posedlinearsystems 28
2.1 Motivation 28
2.2 Definitionsandbasicproperties 34
2.3 Basicexamplesofwell-posedlinearsystems 46
2.4 Timediscretization 55
2.5 Thegrowthbound 60
2.6 Shiftrealizations 67
2.7 TheLax–Phillipsscatteringmodel 71
2.8 TheWeissnotation 76
2.9 Comments 78
3 Stronglycontinuoussemigroups 85
3.1 Normcontinuoussemigroups 85
3.2 ThegeneratorofaC semigroup 87
0
3.3 Thespectraofsomegenerators 98
3.4 Whichoperatorsaregenerators? 106
3.5 Thedualsemigroup 113
3.6 Theriggedspacesinducedbythegenerator 122
3.7 Approximationsofthesemigroup 128
3.8 ThenonhomogeneousCauchyproblem 133
3.9 Symboliccalculusandfractionalpowers 140
3.10 Analyticsemigroupsandsectorialoperators 150
v
vi Contents
3.11 Spectrumdeterminedgrowth 164
3.12 TheLaplacetransformandthefrequencydomain 169
3.13 Shiftsemigroupsinthefrequencydomain 177
3.14 Invariantsubspacesandspectralprojections 180
3.15 Comments 191
4 Thegeneratorsofawell-posedlinearsystem 194
4.1 Introduction 194
4.2 Thecontroloperator 196
4.3 Differentialrepresentationsofthestate 202
4.4 Theobservationoperator 213
4.5 Thefeedthroughoperator 219
4.6 Thetransferfunctionandthesystemnode 227
4.7 Operatornodes 238
4.8 Examplesofgenerators 256
4.9 Diagonalandnormalsystems 260
4.10 Decompositionsofsystems 266
4.11 Comments 273
5 Compatibleandregularsystems 276
5.1 Compatiblesystems 276
5.2 Boundarycontrolsystems 284
5.3 Approximationsoftheidentityinthestatespace 295
5.4 Extendedobservationoperators 302
5.5 Extendedobservation/feedthroughoperators 313
5.6 Regularsystems 317
5.7 Examplesofregularsystems 325
5.8 Comments 329
6 Anti-causal,dual,andinvertedsystems 332
6.1 Anti-causalsystems 332
6.2 Thedualsystem 337
6.3 Flow-inversion 349
6.4 Time-inversion 368
6.5 Time-flow-inversion 378
6.6 Partialflow-inversion 386
6.7 Comments 400
7 Feedback 403
7.1 Staticoutputfeedback 403
7.2 Additionalfeedbackconnections 413
7.3 Statefeedbackandoutputinjection 422
7.4 Theclosed-loopgenerators 425
Contents vii
7.5 Regularityoftheclosed-loopsystem 433
7.6 Thedualoftheclosed-loopsystem 436
7.7 Examples 436
7.8 Comments 440
8 Stabilizationanddetection 443
8.1 Stability 443
8.2 Stabilizabilityanddetectability 453
8.3 Coprimefractionsandfactorizations 465
8.4 Coprimestabilizationanddetection 473
8.5 Dynamicstabilization 485
8.6 Comments 502
9 Realizations 505
9.1 Minimalrealizations 505
9.2 Pseudo-similarityofminimalrealizations 511
9.3 RealizationsbasedonfactorizationsoftheHankeloperator 517
9.4 Exactcontrollabilityandobservability 521
9.5 Normalizedandbalancedrealizations 530
9.6 Resolventtestsforcontrollabilityandobservability 538
9.7 Modalcontrollabilityandobservability 546
9.8 Spectralminimality 549
9.9 Controllabilityandobservabilityoftransformedsystems 551
9.10 Timedomaintestsandduality 554
9.11 Comments 565
10 Admissibility 569
10.1 Introductiontoadmissibility 569
10.2 Admissibilityandduality 572
10.3 ThePaley–Wienertheoremand H∞ 576
10.4 Controllabilityandobservabilitygramians 583
10.5 Carlesonmeasures 591
10.6 Admissiblecontrolandobservationoperatorsfordiagonal
andnormalsemigroups 598
10.7 Admissiblecontrolandobservationoperatorsfor
contractionsemigroups 602
10.8 AdmissibilityresultsbasedontheLax–Phillipsmodel 610
10.9 Comments 613
11 Passiveandconservativescatteringsystems 616
11.1 Passivesystems 616
11.2 Energypreservingandconservativesystems 628
11.3 Semi-losslessandlosslesssystems 636
viii Contents
11.4 Isometricandunitarydilationsofcontractionsemigroups 643
11.5 Energypreservingandconservativeextensionsof
passivesystems 655
11.6 Theuniversalmodelofacontractionsemigroup 660
11.7 Conservativerealizations 670
11.8 Energypreservingandpassiverealizations 677
11.9 TheSpectrumofaconservativesystem 691
11.10 Comments 692
12 Discretetimesystems 696
12.1 Discretetimesystems 696
12.2 Theinternallinearfractionaltransform 703
12.3 TheCayleyandLaguerretransforms 707
12.4 Thereciprocaltransform 719
12.5 Comments 728
Appendix 730
A.1 Regulatedfunctions 730
A.2 Thepositivesquarerootandthepolardecomposition 733
A.3 Convolutions 736
A.4 Inversionofblockmatrices 744
Bibliography 750
Index 767