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�� "!#�%$&��'�'� �'(��*),+"��'�.-/+��'� ��10 �2�&�2+3�4��506��7��3�8��9;:=KJLG8M�:=JLG�MN? ?

Kalle Mikk ola: Infinite-Dimensional Linear Systems, Optimal Control and AlgebraicRiccati Equations; Helsinki University of Technology Institute of MathematicsResearchReports A452 (2002).

Abstract: In this monograph, we solve rather general linear, infinite-dimensional,time-invariant control problems,including the H∞ and LQR prob-lems,in termsof algebraic Riccatiequationsandof spectral or coprimefactoriza-tions.Wework in theclassof (weaklyregular) well-posedlinear systems(WPLSs)in thesenseof G. WeissandD. Salamon.Moreover, we develop the required theories,also of independentinterest, onWPLSs,time-invariant operators,transfer andboundary functions,factorizationsandRiccatiequations. Finally, wepresentthecorresponding theoriesandresultsalsofor discrete-timesystems.

AMS subject classifications: 42A45, 46E40,46G12, 47A68, 49J27, 49N10, 49N35,93-02,93A10,93B36,93B52,93C05,93C55,93D15

Keywords: suboptimal H-infinity control, standard H-infinity problem, measurementfeedbackproblem; H-infinity full information control problem, state feedbackproblem;Nehari problem; LQC, LQR control, H2 problem, minimization; bounded real lemma,positive real lemma; dynamic stabilization, controller with internal loop, strong stabi-lization, exponential stabilization, optimizability; canonical factorization, (J,S)-spectralfactorization, (J,S)-inner coprime factorization, generalizedfactorization, J-losslessfac-torization; compatible operators,weakly regular well-posedlinear systems, distributedparametersystems;continuous-time,discrete-time; infinite-horizon; time-invariant oper-ators, Toeplitz operators, Wiener class, equally-spaceddelays, Popov function; transferfunctions, H-infinity boundary functions, Fourier multipliers; Bochnerintegral, stronglymeasurablefunctions, strong Lp spaces; Laplacetransform,Fouriertransform

[email protected]

ISBN 951-22-6153-7ISSN0784-3143Espoo, 2002

Helsinki University of TechnologyDepartmentof Engineering PhysicsandMathematicsInstitute of MathematicsP.O.Box 1100, 02015 HUT, Finland

email:[email protected] http://www.math.hut.fi/

Contents

Volume1/31 Intr oduction 11

1.1 Onthecontributionsof thisbook . . . . . . . . . . . . . . . . . . 121.2 A summaryof thisbook . . . . . . . . . . . . . . . . . . . . . . 151.3 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

I TI Operator Theory 45

2 TI and MTI Operators 472.1 Time-invariantoperators(TI) . . . . . . . . . . . . . . . . . . . . 482.2 " TIC — invertibility . . . . . . . . . . . . . . . . . . . . . . . . 552.3 Staticoperators . . . . . . . . . . . . . . . . . . . . . . . . . . . 632.4 Thesignature operatorS . . . . . . . . . . . . . . . . . . . . . . 652.5 Losslessness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682.6 MTI andits subclasses. . . . . . . . . . . . . . . . . . . . . . . 71

3 Transfer Functions ( #TI $ L∞strong, %TIC $ H∞) 793.1 Transferfunctionsof TI ( #TI $ L∞strong) . . . . . . . . . . . . . . . 803.2 #TI $ L∞strongfor Banachspaces(FourierMultipliers) . . . . . . . . 903.3 H2 andH∞ boundaryfunctionsin L2 andL∞strong . . . . . . . . . . 99

4 CoronaTheoremsand Inverses 115

5 Spectral Factorization ( &'$)(+*-, , ./* J .0$0,+* S, ) 1335.1 Auxili ary spectralfactorizationresults . . . . . . . . . . . . . . . 1335.2 MTI spectralfactorization( &21�(31�,54 MTI) . . . . . . . . . . . . 140

II Continuous-TimeControl Theory 149

6 Well-PosedLinear Systems(WPLS) 1516.1 WPLStheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1536.2 Regularity ( 6 #.87 À ∞ 9 ) . . . . . . . . . . . . . . . . . . . . . . . . 1666.3 Furtherregularityandcompatibility . . . . . . . . . . . . . . . . 1806.4 Spectralandcoprimefactorizations( .:$0;= 1) . . . . . . . . . 2026.5 Furthercoprimenessandfactorizations. . . . . . . . . . . . . . . 210

3

4 CONTENTS

6.6 FeedbackandStabilization(ΣL, Σ @ , Σ A ) . . . . . . . . . . . . . . . 2196.7 Furtherfeedbackresults. . . . . . . . . . . . . . . . . . . . . . . 2446.8 Systemswith B Bu0 4 Lp 7�C 0 1 1D ;H 9 . . . . . . . . . . . . . . . . . 2636.9 BoundedB, boundedC, PS-systems. . . . . . . . . . . . . . . . 272

7 Dynamic Stabilization 279

7.1 Dynamicfeedback(DF) stabilization ( 7 I EGF 0H2I0 J 9K> 1 4 TIC) . . . 2807.2 DF-stabilizationwith internalloop . . . . . . . . . . . . . . . . . 2917.3 DPF-stabilization( LNMO7P.�1�Q39 ) . . . . . . . . . . . . . . . . . . . . 314

Volume2/3

III Riccati equationsand Optimal control 347

8 Optimal Control ( ddu R $ 0) 3498.1 AbstractJ-critical control(Jyycrit S ∆y) . . . . . . . . . . . . . . 3518.2 AbstractJ-coercivity (RUTV C u DuD ) . . . . . . . . . . . . . . . . . 3568.3 J-critical controlfor WPLSs . . . . . . . . . . . . . . . . . . . . 3618.4 J-coercivity andfactorizations . . . . . . . . . . . . . . . . . . . 3778.5 Problemsonafinite time interval . . . . . . . . . . . . . . . . . . 3928.6 Extendedlinearsystems(ELS) . . . . . . . . . . . . . . . . . . . 395

9 Riccati Equationsand J-Critical Control 4019.1 TheRiccatiEquation:A summaryfor W out (r.c.f.X CARE) . . . . 4049.2 Riccatiequationswhen B Bu0 4 L1 . . . . . . . . . . . . . . . . . 4179.3 Proofsfor Section9.2 . . . . . . . . . . . . . . . . . . . . . . . . 4329.4 Analytic semigroups . . . . . . . . . . . . . . . . . . . . . . . . 4389.5 ParabolicproblemsandCAREs . . . . . . . . . . . . . . . . . . 4439.6 Parabolicproblems:proofs . . . . . . . . . . . . . . . . . . . . . 4509.7 RiccatiequationsonDom7 Acrit 9 . . . . . . . . . . . . . . . . . . 4529.8 Algebraicandintegral Riccatiequations(CAREX IARE) . . . . . 4659.9 J-Critical control X RiccatiEquation . . . . . . . . . . . . . . . 4819.10 Proofsfor Section9.9: Crit X eIARE . . . . . . . . . . . . . . . . 5009.11 Proofsfor Section9.8: eCAREX eIARE . . . . . . . . . . . . . . 5079.12 FurthereIARE andeCAREresults . . . . . . . . . . . . . . . . . 5179.13 Examplesof Riccatiequations . . . . . . . . . . . . . . . . . . . 5259.14 7 J 1�YZ9 -critical factorization( .:$0;= 1) . . . . . . . . . . . . . . 5349.15 H2-factorizationwhendimU [ ∞ . . . . . . . . . . . . . . . . . 539

10 Quadratic Minimization (minR ) 54310.1 Minimizing \ ∞0 7�] y ] 2H À ] u ] 2U 9 dm (LQR) . . . . . . . . . . . . . 54510.2 Generalminimization(LQR) . . . . . . . . . . . . . . . . . . . . 55310.3 Standardassumptions . . . . . . . . . . . . . . . . . . . . . . . . 56810.4 TheH2 problem. . . . . . . . . . . . . . . . . . . . . . . . . . . 58010.5 Reallemmas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587

CONTENTS 5

10.6 PositivePopov operators(0 ^_.�* J .0$),+*O, ) . . . . . . . . . . . 59210.7 PositiveRiccatiequations(R 7 0 1�`a9=b 0) . . . . . . . . . . . . . . . 601

11 H∞ Full-Inf ormation Control Problem( ] w TV z ]2[ γ) 60711.1 TheH∞ Full-Info ControlProblem(FICP) . . . . . . . . . . . . . 60811.2 TheH∞ FICP:proofs . . . . . . . . . . . . . . . . . . . . . . . . 62511.3 TheH∞ FICP:stablecase. . . . . . . . . . . . . . . . . . . . . . 64911.4 MinimaxJ-coercivity . . . . . . . . . . . . . . . . . . . . . . . . 66511.5 Thediscrete-timeH∞ ficp . . . . . . . . . . . . . . . . . . . . . . 66911.6 TheH∞ ficp: proofs . . . . . . . . . . . . . . . . . . . . . . . . . 67311.7 TheabstractH∞ FICP . . . . . . . . . . . . . . . . . . . . . . . . 67611.8 TheNehariproblem. . . . . . . . . . . . . . . . . . . . . . . . . 68011.9 Theproofsfor Section11.8 . . . . . . . . . . . . . . . . . . . . . 682

12 H∞ Four-Block Problem( ]�LcMO7P.�1�Q39d]2[ γ) 68512.1 ThestandardH∞ problem(H∞ 4BP) . . . . . . . . . . . . . . . . 68612.2 Thediscrete-timeH∞ problem(H∞ 4bp) . . . . . . . . . . . . . . 70512.3 Thefrequency-space(I/O) H∞ 4BP . . . . . . . . . . . . . . . . . 70912.4 Proofsfor Section12.3 . . . . . . . . . . . . . . . . . . . . . . . 71912.5 Proofsfor Section12.1— 4BP e X 1fe Y 1fe Z . . . . . . . . . . . . . 73312.6 Proofsfor Section12.2— 4bp e X 1fe Y 1fe Z . . . . . . . . . . . . . 758

Volume3/3

IV Discrete-Time Control Theory (wpls’s) 775

13 Discrete-Time Maps and Systems(ti & wpls) 77713.1 Discrete-timeI/O maps(tic) . . . . . . . . . . . . . . . . . . . . 77913.2 TheCayley transform( g , h ) . . . . . . . . . . . . . . . . . . . . 78713.3 Discrete-timesystems(wpls7 U 1 H 1 Y 9 ) . . . . . . . . . . . . . . . 79213.4 Time discretization (∆S : WPLS V wpls) . . . . . . . . . . . . . . 805

14 Riccati Equations (DARE) 81514.1 Discrete-timeRiccatiequations(DARE) . . . . . . . . . . . . . . 81514.2 DARE — furtherresults . . . . . . . . . . . . . . . . . . . . . . 82014.3 Spectralandcoprimefactorizations. . . . . . . . . . . . . . . . . 829

15 Quadratic Minimization 83315.1 J-critical controlandminimization . . . . . . . . . . . . . . . . . 83315.2 Standardassumptionsin discretetime . . . . . . . . . . . . . . . 83915.3 PositiveDAREs . . . . . . . . . . . . . . . . . . . . . . . . . . . 84215.4 Reallemmas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84315.5 Riccati inequalitiesandthemaximal solution . . . . . . . . . . . 846

Conclusions 851

6 CONTENTS

A Algebraic and Functional Analytic Results 853

A.1 Algebraicauxiliary results( F A110 A12A22J > 1 $ji A k 1110 > A k 111 A12A k 122A k 122 l ) . . . . 854A.2 Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . 863A.3 Hilbert andBanachspaces . . . . . . . . . . . . . . . . . . . . . 865A.4 C0-Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . 896

B Integration and Differ entiation in BanachSpaces 903B.1 TheLebesgueintegral andL p 7 R; C 0 1 À ∞ DP9 spaces . . . . . . . . . 904B.2 Bochnermeasurability( f 4 L 7 Q;B9 ) . . . . . . . . . . . . . . . . 906B.3 Lebesguespaces(Lp 7 Q 1 µ;B9 ) . . . . . . . . . . . . . . . . . . . 912B.4 TheBochnerintegral ( \ Q : L1 7 Q;B9 V B) . . . . . . . . . . . . . 922B.5 Differentiationof integrals( ddt \ ) . . . . . . . . . . . . . . . . . . 937B.6 Vector-valueddistributions monp7 Ω;B9 . . . . . . . . . . . . . . . . 943B.7 Sobolev spacesWk q p 7 Ω;B9 . . . . . . . . . . . . . . . . . . . . . 945

C Almost Periodic Functions (AP) 953

D Laplaceand Fourier Transforms ( r s u $ #u) 957E Inter polation Theorems 983

E.1 Interpolationtheorems(L p1À

Lp2 V Lq1 À Lq2) . . . . . . . . . . 983F Lpstrong, L

pweakand Integration 993

F.1 LpstrongandLpweak . . . . . . . . . . . . . . . . . . . . . . . . . . 993

F.2 Strongandweakintegration(s\ 1 w\ ) . . . . . . . . . . . . . . . . . 1007F.3 WeakLaplacetransform( r s w) . . . . . . . . . . . . . . . . . . . 1013

References 1020

Notation 1033Glossary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046

Index 1047

List of Figures

Volume1/3

1.1 Input/state/outputdiagramof aWPLS Fut vcwI J . . . . . . . . . . . . 231.2 Dynamicoutputfeedbackcontroller Q for .54 TIC∞ 7 U 1 Y 9 . . . . 251.3 TheH∞ FICP . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

1.4 DPF-controllerxΣ for Σ 4 WPLS7 U y W 1 H 1 Z y Y 9 . . . . . . . . 376.1 Input/state/outputdiagramof thesystemΣ . . . . . . . . . . . . . 1556.2 Staticoutputfeedback. . . . . . . . . . . . . . . . . . . . . . . . 221

6.3 Statefeedbackconnection . . . . . . . . . . . . . . . . . . . . . 225

6.4 Thesettingof Proposition6.6.18(f). . . . . . . . . . . . . . . . . 233

6.5 Outputinjectionconnection. . . . . . . . . . . . . . . . . . . . . 240

6.6 TheextendedsystemΣTotal . . . . . . . . . . . . . . . . . . . . . 2416.7 Theclosed-loopsystem7 ΣTotal 9 L . . . . . . . . . . . . . . . . . . 2416.8 Thesettingof Lemma6.7.11(a’) . . . . . . . . . . . . . . . . . . 254

6.9 Thesettingof Lemma6.7.12 . . . . . . . . . . . . . . . . . . . . 259

7.1 DF-controllerQ for .54 TIC∞ 7 U 1 Y 9 . . . . . . . . . . . . . . . . 2837.2 DF-controllerxΣ for Σ 4 WPLS7 U 1 H 1 Y 9 . . . . . . . . . . . . . . 2847.3 DF-controllerz with internalloop for .54 TIC∞ 7 U 1 Y 9 . . . . . . 2937.4 DF-controllerxΣ with internalloop for Σ 4 WPLS7 U 1 H 1 Y 9 . . . . 2957.5 Controller (2,{> 1 with r.c. internalloop . . . . . . . . . . . . . . . 3077.6 Controller x, > 1 x( with l.c. internalloop . . . . . . . . . . . . . . . 3077.7 Thecontroller Q n : $)(|7}, À E (�9 > 1 : xy TV u for . À E . . . . . . . 3147.8 DPF-controllerQ for .54 TIC∞ 7 U y W 1 Z y Y 9 . . . . . . . . . . 3177.9 DPF-controllerxΣ for Σ 4 WPLS7 U y W 1 H 1 Z y Y 9 . . . . . . . . 3197.10 DPF-controllerz with internalloop for .54 TIC∞ 7 U y W 1 Z y Y 9 3197.11 DPF-controllerxΣ with internalloopfor Σ 4 WPLS7 U y W1 H 1 Z y Y 9 3207.12 Thecontroller Q n : $)(|7}, À E (�9 > 1 : xy TV u for . À F 0E 00J . . . . . 340

7

8 LIST OF FIGURES

Volume2/3

9.1 Statefeedbackconnection . . . . . . . . . . . . . . . . . . . . . 4089.2 Thesettingof Lemma9.12.3 . . . . . . . . . . . . . . . . . . . . 5259.3 Thesettingof Proposition9.12.4 . . . . . . . . . . . . . . . . . . 525

10.1 TheH2 problem. . . . . . . . . . . . . . . . . . . . . . . . . . . 58410.2 TheH2 statefeedbackproblem . . . . . . . . . . . . . . . . . . . 586

11.1 A WPLScontrolledby aH∞-FI-pair . . . . . . . . . . . . . . . . 61511.2 Dynamicfeedforwardcompensator. . . . . . . . . . . . . . . . . 65811.3 Parameterizationof all suboptimal compensators . . . . . . . . . 658

12.1 Thesuboptimal controller Q : $)L/MO7f~219 : y TV u . . . . . . . . . 69912.2 Themap ./ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72112.3 “DPF-controller Q with internal loop” for Σ 4 WPLS7 U y

W1 H 1 Z y Y 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742

Preface

For H∞, H2, LQR andseveralotherlinear time-invariant controlproblems,itis well known thattheexistencesof

(I) asolution of thatcontrolproblem,(II) acorrespondingcoprimeor spectralfactorizationand(III) a stabilizing solution of thecorrespondingRiccatiequation

are, roughly speaking,all equivalent in the finite-dimensional setting,andfromoneof themthe otherscanbe computed.Therefore,control problemsareoftensolvedby computingthesolutionsof thecorrespondingRiccatiequations.

Theseresultshave beenextendedto infinite-dimensional(semigroup)controlsystemswith boundedinput andoutputoperators,and in the eightiesandearlyninetiesalsoto thelargerclassof Pritchard–Salamonsystemsandto certainotherspecialcasesof oursetting.

Themainpurposeof this monographis to generalizetheseresultsto infinite-dimensional (weaklyregular)well-posedlinearsystems(WPLSs)in thesenseofG. Weiss. This is donein Chapters8–12; seepp. 21–24for an introduction toWPLSs.

We also develop correspondingdiscrete-timeresults(Chapters13–15 andSections11.5and12.2),WPLS theory(Chapters6–7, includingregularity, statefeedback,output injection and dynamic feedback),and an extensive theoryof independentinterestfor time-invariant operators(“Toeplitz operators”)andsomeof their subclasses(such as extendedCallier–Desoerclassesand otherconvolutions with measures),transferandboundaryfunctionsandspectralandcoprime factorization(Chapters2–5 and Sections6.4–6.5). A more detaileddescriptionof someof the main resultsof this monographandsomehistoricalremarksare provided in Sections1.1 and 1.2 and in the “notes” partsof eachsection.

WPLSscover all linear time-invariant systems that mapthe initial stateandinputcontinuously to thestateandoutput(with inputsandoutputs in L 2loc; seepp.21); in particularall settingsmentionedabovearecovered.Moreover, any transferfunction that is boundedandholomorphic on someright half-plane(i.e., that iswell-posedor proper) hasa WPLSrealization.Theinputandoutputoperatorsofa WPLS maybeasunboundedasfor Pritchard–Salamonsystems independentlyof eachotheraslongasthetransferfunctionis well posed,thusallowing roughlytwiceasmuchirregularity.

Weakregularitymeanstheexistenceof a feedthroughoperatorin averyweaksense;an equivalentconditionis that the transferfunctionhasa limit at infinityalongthepositive realaxis. In particular, all I/O mapswhoseimpulse responseisa (uniform, strongor weak)L p functionplussomedelays(or any vector-valuedmeasure)andseveralothersareweaklyregular.

Much of our theoryon WPLSscover thegeneralcase,but Riccati equationscannotbedefinedwithout feedthroughoperators(exceptin a veryweaksense,asin Section9.7).

We generallyallow theinput,stateandoutputspacesof WPLSsto beHilbertspacesof arbitrarydimensions,andsomeresultsaregivenevenin aBanachspace

setting. In addition to exponentially stabilizing controllersand exponentiallystabilizingsolutions of Riccati equations,we alsostudystabilizing andstronglystabilizing ones; part of theseresultsmay be new even for finite-dimensionalsystems.

During the last four decades,the literaturehasbecomeabundantin infinite-dimensional(or distributed) systemsarisingin physics, engineering,economics,mechanics,environmentalmodeling, biomedicalengineering,evolution dynam-ics, geophysicsandothersciences,and the systems can representsemiconduc-tor devices,animalpopulations,fluid dynamics,microwave circuits,vibrationofstringsor membranes,heatdiffusion,computerharddiscs,CD playersandmanyotherdevices.

For some particular systemsand problems, there are now rather maturetheories.Thepurposeof thismonographis tosolvetheproblemsin averygeneral,unifying framework — in theframework of WPLSs.

Our presentationis abstractand theory-oriented;nevertheless,many of ourresultscanbeunderstoodwithout thefunctionalanalyticknowledgeprovidedbytheappendices.Thebookis ratherself-containedandit canbereadwithout anyprior knowledgein systemor control theory, althoughexpertsareconsideredasthemainaudienceandsomeproofsmaybedemanding.

Acknowledgements: I am very grateful to professorOlof Staffans, whowarmly guided me to the world of mathematical control theory, and whoseknowledgehasbeenveryvaluableto me.I alsowish to thankprofessorStig-OlofLondenfor helpingme not get lost in Fréchetspacesduring my undergraduatestudies,and my supervisor, professorOlavi Nevanlinna, and the peopleat theInstituteof Mathematics at Helsinki University of Technologyfor its warm andstimulatingenvironment.

ProfessorIlya Spitkovsky haskindly pointedme to several importantresultsonspectralfactorizationandIrenaLasieckaandRobertoTriggiani to many resultsobtainedby their school. I have alsohad inspiring discussionswith professorsRuthCurtain,GeorgeWeissandHansZwart,doctorsMarko HuhtanenandJarmoMalinen, and others. Part of this work was written with the supportof theAcademyof Sciences,theGraduateSchoolof AnalysisandtheFinnishCulturalFoundation.Finally, I wish to thankmy friendsandloved ones,for makingthisworld suchagreatplaceto live in.

Kalle Mikkola

Twiceor thricehadI lovedtheeBefore I knew thy faceor name.Soin a voice, soin a shapelessflame,Angelsaffectusoft, andworshippedbe.

— JohnDonne(1571–1631)

Chapter 1

Intr oduction

Fromthewreck of thepast,which hathperish’d,Thusmuch I at leastmayrecall,It hathtaughtmethatwhatI mostcherish’dDeservedto bedearestof all.

— Lord Byron (1788–1824),"Stanzasto Augusta"

In Section1.1, we summarize the main contributions of this monograph,avoiding any technicalities. Readerswishing to get a somewhat moreaccuratepictureon theactualresultsshouldconsultSection1.2,wherewegiveaglanceateachchapterby explaining its contentsbut yetavoidingmosttechnicaldetailsandgenerality.

Someconventionson notation,proofsandhypothesesareexplainedin Sec-tion 1.3.Seetheendof thebookfor symbols,concepts,abbreviations,referencesandindex.

11

12 CHAPTER1. INTRODUCTION

1.1 On the contributions of this book

Our ultimate goal hasbeento develop the H∞ Four-Block Problemtheory inChapter12. This has requiredus to first develop several other parts of thetheory that areof independentinterest,suchasthe Riccati equationtheory, thecostminimizationtheory, thedynamicfeedbacktheory, theWPLS theoryor thediscrete-timetheory, all of whicharemainlygeneralizationsof existingtheoryforfinite-dimensionalor smoothinfinite-dimensionalsystems.

Our mainresultsincludethefollowing:

1. On (generalized)Optimal Control and Riccati equationsfor WPLSs,wehave

(a) establishedtherelationsbetweendifferentclassicalcoercivity assump-tions(Section10.3),generalizedthemto WPLSsandappliedthemtosolve thegeneralcontrolproblem(Section8.4).

(b) formulatedIntegral AlgebraicRiccati Equationsto establishthe cor-respondingequivalencein continuous time. This alsoallowed us toreduceseveralproblemsto discretetime, whereinput andoutputop-eratorsarebounded.

(c) established the corresponding equivalence for (classical-type)Continuous-timeAlgebraicRiccatiEquations(underweakregularity)(Chapter9).

i. The implication from the existenceof a solution of the controlproblemto theexistenceof asolution of theRiccatiEquationwasalreadyestablishedby G. Weiss,M. WeissandO. Staffansunderstrongerregularityandverystrongstabilizability anddetectabilityassumptions.

ii. We have also shown the existenceof a smoothersolution un-derseveraldifferentadditional regularity assumptions (e.g.,Sec-tion 9.2).

(d) establishedthe Continuous-time Riccati equationson the domainofthe closed-loopsemigroupgeneratorfor general(possiblyirregular)WPLSs(Section9.7;extensionof [FLT]).

(e) treatedall the above for both the exponentially stabilizing controlsandfor therecently-popularstronglyor output-stabilizingcontrolsandothers,thusprovidingnew resultsevenfor finite-dimensionalsystems.

2. On specific control problemsfor WPLSs, we have extendedthe finite-dimensional resultsby, in addition to theabove,solving

(a) the H∞ full-information control problem in terms of the Riccatiequation(Chapter11).

1.1. ON THE CONTRIBUTIONS OF THIS BOOK 13

i. In thestablecase,theexistenceof a solutionwasalreadyshownby O. Staffans,assumingtheexistenceof correspondingspectralfactorization;a similar statementappliesto the LQR problembelow.

(b) the general(measurementfeedback,or four-block) H∞ controlprob-lem in termsof two Riccatiequationsanda spectralradiuscondition(Chapter12). Wehaveshown thattherecenttheoryof controllerswithan internalloop (cf. [CWW96]) is shown to be intimately connectedto a generalsolution of this problem,andall suchsolutions arealsocovered.

(c) thecostminimization(LQR) problem,showing theexistenceof a so-lution equivalentto theexistenceof any solution of thecorrespondingRiccati equation(the solution neednot be stabilizing or even admis-siblea priori). We have alsoderivedsimilar generalizationsof StrictBoundedandStrictly Positive(Real) Lemmas(Chapter10).

3. On WPLSsystemtheory, wehave

(a) introducedcompatibility, which allows oneto write any WPLS in adifferentialform regardlessof regularity (Section6.3).

(b) introducedaninfinite-dimensional weaklycoprimefactorizationcon-cept(Sections6.4and6.5)andappliedit to establishthestability anduniquenessof asolution of certainRiccatiequationsandcontrolprob-lems(this is particularlyusefulwhen the solution is not requiredtobeexponentially stabilizing). This conceptandcompatibility have al-readybecomethesubjectsof leadingresearchers’ articles.

(c) characterizedthe transferfunctions(equivalently, impulseresponses)having a Pritchard–Salamonrealization(thuscorrectingtheerrorsin[KMR] to which we alsoprovide a counter-example). Similarly, wehavecharacterizedtransferfunctionsrealizablewith boundedinputoroutputoperators.(Section6.9)

(d) generalizedtheequivalencebetweenexponential dynamicstabilizabil-ity andexponential stabilizability anddetectability(Theorem7.2.4).

4. The infinite-dimensional control theoryhasbeenlimited by several openproblemsin harmonicandfunctionalanalysisandfunctiontheory. Thishasleadusto solve thosemostintimatelyconnectedto ourwork, e.g.,wehave

(a) generalizedtheL2 Fouriermultiplier theoremto thecaseof functionswith valuesin Hilbert spaces(theseparablecasewasalreadyknown)andbeyond(Theorem1.2.2).

(b) generalizedsimilarly theexistenceresultof theboundaryfunctionofaH∞ function(Theorem1.2.3).


(c) developed a theory of strongly measurableoperatorvalued func-tions, including the the completenessof L∞strong (and incompletenessof Lpstrong) and its applicationsincluding the two above results(Ap-pendixF).

(d) shown the existenceof a spectralfactorizationfor convolutions with(Hilbert space)operator-valuedmeasureshaving a discretepart plusan L1 part (assuming the invertibility of the Toeplitz operator;seeTheorems1.2.4and1.2.5).

(e) extendedto the infinite-dimensional casethe classical[ClaGoh] H2

spectralfactorizationfor any Popov function having an invertibleToeplitzoperator(Theorem9.14.6).

Finally, of all the above we alsopresentcorrespondinginfinite-dimensionaldiscrete-timeresults,which becomeratherelegantsince,in this case,the inputandoutputoperatorsarenaturallybounded.

For a control theorist, the generalizationof Riccati equationtheory to theregular WPLS setting(particularly 1b and 1c above) and the generalH∞ andminimizationproblems(2.) mayriseabovetherest.

To observein detailtheothernew resultsin thismonograph,thereadershouldreadthe“Notes” at theendof eachsection.Therewe discussearlierresearchinsamedirection,including any known similar resultsunderlessgeneralsystems,settingsor assumptions.

The sizeof this book requiressomeexplanations. For the first, the chaptersof this monographareso intimately connectedto eachother that it would havebeenimpossible to removeasinglechapterwithout destroying, e.g.,theproofsinChapter12.

If we hadlimited ourselvesonly to very smoothsystemsor to discrete-timesystems,thesizeof this bookwould have probablyfallenby morethanhalf butits contribution evenby muchmore. Indeed,mostproblemsbut alsomostvaluein our work is in its generality. Certainly, we might have presentedour solutionsonly in termsof factorizations(which aregiven asan intermediarystagein ourproofs),but theRiccatiequationsarereally theform of theclassicalsolutionsandsomethingthatprovidesa practicalway to solve theproblems.

SometimestheRiccatiequationsbecomeverycomplicatedfor generalregularWPLSs,hencewehavepresentedmorebeautifulcorollariesfor importantspecialcases,suchasfor thecasewheretheI/O mapis theconvolution with a measure.Moreover, therealizationof theoptimal controlin theform of astatefeedbackordynamicfeedbackcontrollerrequiresthe existenceof certainfactorizationsthatneednotexist in thegeneralcase(seeExample11.3.7).

One of the objectives of this book hasbeento stateand prove resultsof atechnicalnaturethataretoo long to bepublishedin ordinaryresearcharticlesbutthatarenecessarybuilding blocksfor thefinal results.

1.2. A SUMMARY OF THIS BOOK 15

1.2 A summary of this book

We now start a rather self-containedsummary, aiming to give the readeramotivation for and a picture of the theory treatedin eachchapter, by startingwith a non-technicaldescriptionandthenpresentingsomeresults. We stronglyrecommendfor thereaderto readthesummariesin thissectionbeforediving intothetechnicalitiesof theactualchapters.

The resultsmentioned below are just examples from the theory; here wehave usually favored simple, important examplesto more generalbut morecomplex ones.Seethechaptersthemselvesfor furtherdefinitions,results,details,explanationsandreferences.

Outside the appendices,the lettersH, U , W, Y andZ will denotecomplexHilbert spacesof arbitrarydimensionsunlesssomethingelseis indicated.

Part I: TI Operator Theory

TheappendicesandPart I of thebookcontainresultsin harmonicandfunctionalanalysis(vector-valuedfunctions,shift-invariantoperators,transferfunctionsandboundaryfunctions,theCoronaTheoremandspectralfactorizationamongothers)thatareneededin thecontroltheoryof PartsII–IV. Many of theresultsarealsoofindependentinterest.A fasttrackto WPLSsis to first haveaglanceatsubsections2.1.1–2.1.7andthengodirectly to Part II.

Chapter 2: TI and MTI Operators (MTI TI)In Chapter2, we study the theory TIω, the spaceof bounded,shift-invariantoperatorsL2 V L2, where the L2 spacemay have a weight and the functionshave their valuesin a Hilbert space.We alsopresentcertainsmoothsubclassesof TI, particularlyMTI, theconvolutionswith a (vector-valued)measurewith nosingularcontinuouspart.

Our contributions include the theory of the intersectionTIω TIω and itscausalpartfor two weightsω 1 ω n�4 R (see2.1.9–2.1.11and3.1.6),necessaryandsufficient conditions for losslessnessandcertainresultson staticoperatorsandsignatureoperators.

Technically, TIω 7 U 1 Y 9 is thespaceof boundedtime-invariantlinearoperatorsL2ω 7 R;U 9 V L2ω 7 R;Y 9 , whereU andY areHilbert spacesof arbitrarydimensions,ω 4 R, and ] u ] L2ω : $� R e> 2ωt ] u 7 t 9O] 2U dt 1 2 (1.1)for Bochner-measurableu : R V U (thus, L20 $ L2, L2ω : $ eω u 7�`a9 u 4 L2 ).The time-invarianceof .4 TIω meansthat . τ 7 t 9c$ τ 7 t 9. for all t 4 R, whereτ 7 t 9 u : $ u 7�` À t 9 .

Themapsin TICω 7 U 1 Y 9 : $.4 TIω 7 U 1 Y 9 π > . π $ 0 arecalledcausal(or sometimesToeplitz operators);hereπ u : $ χR u andπ > u : $ χR k u for all


functionsu, andχE is thecharacteristicfunctionof asetE. Thefollowing is wellknown:

Theorem1.2.1 For each .4 TICω 7 U 1 Y 9 , there is a unique function #.4H∞ 7 C ω ; 7 U 1 Y 9�9 , called the transferfunction (or symbol)of . , s.t. . u $ #. ûon C ω for all u 4 L2ω 7 R ;U 9 . Themapping. TV #. is an isometricisomorphismonto.

Here 7 U 1 Y 9 denotesthe space of bounded linear operatorsU V Y,H∞ 7 C ω ; 7 U 1 Y 9�9 denotesthe Banachspaceof boundedholomorphic functionsC ω V 7 U 1 Y 9 , and #u denotestheLaplacetransform#u 7 s9 : $5

Re> stu 7 t 9 dt 7 s 4 C ω : $? s 4 C Res ω 9 (1.2)

of u. Thus,theelementsof TIC∞ 7 U 1 Y 9 : $ ω RTICω 7 U 1 Y 9 correspondone-to-oneto theholomorphic 7 U 1 Y 9 -valuedfunctionsthatareboundedonsomerighthalf-plane;suchfunctionsaregenerallycalled“proper” or “well-posed”. Thesetof theI/O mapsof WPLSsis exactly TIC∞ (seeSection6.1). Transferfunctionsarestudiedalsoin Chapter3.

In Section2.2, we study the invertibility of TICω (and TIω) operators. InSection2.3,we develop sufficient conditions for a TIC operatorto bestatic,thatis, the multiplication operatorinducedby an elementof 7 U 1 Y 9 . We alsogivecertain resultsthat will be usedin connectionwith the signature operators ofoptimization problems,Riccatiequationsandspectralfactorizations.

Also Section2.4treatssignatureoperators.A mainresultof thissectionis thatfor any S 47 U y Y 9 , thefollowing areequivalent:

(i) S $U&*{F IU 00 > IY J & for some&)4 " TIC 7 U y Y 9 ;(ii) S $ E * F IU 00 > IY J E for someE 4 "�7 U y Y 9 .

(Recall that " denotesthesubsetof invertibleoperators.)Section 2.5 treats the concept “ 7 J 1 S9 -losslessness”(close to “ 7 J 1 S9 -

dissipativity”), which is often studiedin connectionwith H∞ problemsand in-definiteinnerproducts(losslessnessis roughlyequivalent to thenonnegativity ofthe correspondingRiccati operator). Thereare two widely-useddefinitionsoflosslessnesswhoseexact connectionhasbeenunknown. We develop necessaryand/orsufficient conditions for bothconceptsandshow that they coincidewhentheinputspacesarefinite-dimensional.

In Section2.6,wedefinethesubclassMTI 7 U 1 Y 9 (“M” for “measures”)astheoperators.54 TI 7 U 1 Y 9 thatareof theform7¡& u 9O7 t 9¢$ ∞∑

k £ 0Tku 7 t E tk 9 À ∞> ∞ f 7 t E r 9 u 7 r 9 dr 1 (1.3)i.e., of the form & u $ µ Y u, where the measureµ consistsof a function f 4L1 7 R; 7 U 1 Y 9�9 plus a discretepart with Tk 4¤7 U 1 Y 9 and tk 4 R for all k 4 N,s.t. ]�&�] MTI : $¥] f ] L1 À ∑

k N ] Tk ]�¦§ U qY ¨ [ ∞ © (1.4)


TheWienerclassMTIL1

refersto theelementsof MTI of form u TV Tu0 À f Y u(i.e., no delays).TheclassMTIC : $ MTI TIC (resp.MTICL1 : $ MTIL1 TIC)consistsof thoseelementsof MTI (resp.MTICL

1) that correspondto measures

supportedonR . In [CD80] and[CZ] amongothers,theclassMTIC (or “ ª7 09 ”)hasbeenstudiedfor finite-dimensionalU andY.

Thebasicpropertiesof theseclassesarelistedin Section2.6.They sharemostpropertiesof mapswith rational transferfunctions; in particular, they have thesamespectralfactorizationproperties(seeSection5.2).Thesepropertiesallow usto show (in Part III) thatclassicalconditionsfor thesolvability of standardcontrolproblemsarenecessaryandsufficientalsofor systemswhoseI/O mapsbelongtoMTIC (suchconditionsaresufficientbut notnecessaryfor generalWPLSs);someof this hasalreadybeenestablishedfor lessgeneralsystems (see,e.g.,[CD80] or[CW99]).

Chapter 3: Transfer Functions ( «TI ¬ L∞strong, TIC ¬ H∞)We studytheLaplaceandFourier transforms(or transferfunctionsor symbols)of TI andTIC maps,thatis, (causalandgeneral)time-invariantmapsL2 V L2.

Our main resultsaretwo generalizationsto unseparableHilbert spaces,firstoneof theFouriermultiplier theorem(“ #TI 7 U 1 Y 9®$ L∞strong7 iR; ¯7 U 1 Y 9�9 ”) andthenof the fact that an operator-valuedH∞ function over the right half-planehasaboundaryfunction in strongL∞ on the imaginaryaxis as its “strong pointwiselimit”, in averynaturalsense.

Wefirst show that“ #TI 7 U 1 Y 9¢$ L∞strong7 iR; ¯7 U 1 Y 9�9 ” (Theorem3.1.3(a1)):Theorem 1.2.2 For each &_4 TI 7 U 1 Y 9 , there is a unique (symbol) #&_4L∞strong7 iR; ¯7 U 1 Y 9�9 s.t. #& #u $ & u a.e. for all u 4 L2 7 R;U 9 . Thismapping & TV #&is an isometricisomorphismof TI 7 U 1 Y 9 ontoL∞strong7 iR; ¯7 U 1 Y 9�9 .

(Theseparablecaseof thisclaimis well-known. HereiR is theimaginaryaxis,and #&°4 L∞strong7 iR; ¯7 U 1 Y 9�9 meansthat #& : iR V ¯7 U 1 Y 9 is s.t. #& u0 4 L∞ 7 iR;Y 9for all u0 4 U . It follows that ] #&±] L∞strong : $ sup² u0 ² U ³ 1 ] #& u0 ] ∞ [ ∞, by LemmaF.1.6.)

Then we go on to show that this Fourier transform restricts to an iso-metric isomorphism of TIa 7 U 1 Y 9 TIb 7 U 1 Y 9 onto H∞ 7 Ca q b; 7 U 1 Y 9�9 , whereH∞ 7 Ca q b; 7 U 1 Y 9�9 refersto boundedholomorphic functionsCa q b V 7 U 1 Y 9 andCa q b : $ s 4 C a [ Res [ b , andthat #& #u $´& u on Ca q b (bothsidesof theequa-tion beingholomorphic) for all u 4 L2a 7 R;U 9 L2b 7 R;U 9 .

In Sections3.1 and3.2,we alsogive further resultson theFourier transformandweaker formsof thetwo resultsmentionedabovefor arbitraryBanachspacesU and Y and Lp in place of L2 (and “TI pω” in place of TIω). Thesecan beconsideredasextensionsof thesocalledFourier multiplier theory.

In Section3.3, we establishseveral resultson the boundaryfunctions ofholomorphic functions,the most importantof which is the following (Theorem3.3.1(c1)):


Theorem1.2.3 For each f 4 H∞ 7 C 0 ; 7 U 1 Y 9�9 , there is a boundaryfunctionf0 4 L∞strong7 iR; ¯7 U 1 Y 9�9 s.t. f0u0 is thenontangential limit of f u0 a.e. on iR forall u0 4 U.

(Theseparablecaseof this theoremwasgivenin [Thomas].)As theobservantreaderalready may have guessed,f0 is the Fourier transform of . , where.4 TI 7 U 1 Y 9 is s.t. #.G$ f . This justifies the useof “ #. ” to denoteboth theFouriertransform#.µ4 L∞strong7 ω À iR; 7 U 1 Y 9�9 andthetransferfunction(Laplacetransform) #.54 H∞ 7 C ω ; 7 U 1 Y 9�9 of amap .54 TICω 7 U 1 Y 9 .

Somecounter-examplesaregiven to show thatTheorem1.2.3is not true forgeneralBanachspacesnorwith H2 in placeof H∞.

Wealsogive furtherresultson transferfunctions;theseresultswill beneededfor theWPLStheoryof PartsII andIII.

Chapter 4: CoronaTheoremsand Inverses

In this chapter, we first show that any causalinversesof I/O mapspreservesmoothnessandthenwedothesamefor causalleft inverses(mostof thisconsistsof combinationsof known results). The latter only holdsfor finite-dimensionalinput spaces,but we presentpartial resultson the infinite-dimensional case,onwhichweshalllaterbuild ourquasi-coprimefactorizationtheoryfor WPLSs.

In Theorem4.1.1,we list the following equivalent conditionsfor the invert-ibility of any .?4 xª¶7 U 1 Y 9 , where xª standsfor TIC, MTIC, CTIC or for someoftheir subclassesmentionedabove:

(i) .54 " xª ;(ii) .54 " TIC;(iii) π �. π 4·"�¯7 π L2 9 ;(iv) #.54·" H∞, i.e., #.+> 1 existsandis boundedonC .In particular, xª is inverse-closedin TIC. Thesameholdsfor thesetof maps

that are “exponentially xª ”. For the casedimU $ dimY [ ∞, thereareseveralotherequivalentconditions, suchas(v) infC ¹¸ det7 #.39 ¸ 0; (vi) . is left-invertiblein TIC (seetheCoronaequivalencebelow for more).

We also give analogousresults on TI, MTI, CTI and their (noncausal)subclasses(e.g., &?4 MTI is invertible in MTI if f #& is boundedlyinvertible oniR) andfurtherinvertibility results.

Then we study the Corona Theoremand its consequencesfollowing themethodsof M. Vidyasagar. In case.:4 xª¤7 U 1 Y 9 , dimU [ ∞, welist thefollowingequivalentconditionsfor theleft-invertibility of . :

(i) º2.´$ I for someº4 xª¶7 Y 1 U 9 ;(ii) º».$ I for someº4 TIC 7 Y1 U 9 ;(iii) #.87 s9�* #.87 s9=b εI for all s 4 C andsomeε 0;(iv) ]K. u ] L2ω b ε ] u ] L2ω for all u 4 L2ω 7 R;U 9 , ω 0 andsomeε 0;


(v) ./* π > .5b επ > onL2 for someε 0;(vi) . t * . t b επ ¼ 0 q t ¨ for all t 0 andsomeε 0.

(Here . t : $ π ¼ 0 q t ¨ . π ¼ 0 q t ¨ .) It follows that ;½4 xª¶7 U 1 Y 9 and


Section5.1 consists of ratherstraight-forward derivation of requiredresultsfrom the literature. In Section5.2, we treat the convolutions with measuresconsistingof a discretepart plus an (uniformly measurable)L1 part. Ourmain contribution is Lemma5.2.3, by which we can reducethe factorizationof suchconvolutionsto theseparatefactorizationsof thediscreteandabsolutelycontinuouspartsof themeasure,whichalreadyhavebeengraduallysolvedduringthelastthreedecades.

Thepositivecaseof thelemmahasalreadybeenprovedby J.Winkin [Winkin](for finite-dimensional input and output spaces). Though the existenceof aspectralfactorizationis always guaranteedin the positive case(assuming theinvertibility of the correspondingToeplitz operator),it is importantto know thesmoothnessof thefactor, asexplainedabove.

The main corollariesof our lemma are that such convolutions mapshavespectralfactors,andthat theseareof the sameform asthe original maps. Thisallowsoneto formulatethesolutionsto WPLScontrolproblemsasin theclassicalcase,thoughwith several technicalcomplications dueto the unboundednessofinput andoutputoperators(seeSection9.1). Thesecorollariescanbewritten intheform of thefollowing two theorems:

Theorem1.2.4(PositiveMTI spectral factorization) LetU bea Hilbert space,and let ª be one of the classesTI, MTI , MTI L1. Let &Ì4°ª7 U 9 , and setxª : $Uª TIC.

Then&'Í 0 iff & hasa factorization&'$), * ,¹1 where ,54 " xª7 U 9K© (1.5)Moreover, if &U4oª exp, then ,+Î 1 4 xª exp.

(The classª exp (resp. xª exp) consistsof “exponentiallystableª (resp. ª exp)maps”.By “ &'Í 0” (or “0 ^& ” ) wemeanthat &)b εI for someε 0.)

If &4 MTI $ª , then #& and #, are continuousin iR, hencethen (1.5) isequivalentto “ #&{7 it 9�$ #,87 it 9Ï* #,87 it 9 for all t 4 R, ,¹1Ï,�> 1 4 MTIC 7 U 9 ”.

Thegeneral(indefinite)caseis analogousexceptthatfor someclassesª , ourresultrequiresU to befinite-dimensional:

Theorem1.2.5(MTI spectral factorization) Let &'4Àª7 U 9 , where ª and xª areas in Theorem 5.2.7. Then the Toeplitz operator(or Wiener–Hopf operator)π & π 47 L2 7 R ;U 9�9 is invertibleiff & hasa spectral factorization&Ð$0( * ,¹1 where ,¹1�(54·" xª¶7 U 9K© (1.6)

Moreover, if &)4ª exp, then ,+Î 1 1p(+Î 1 4 xª exp. (Notethatπ & π 4o7 L2 7 R ;U 9�9 if f & π À π > 4o7 L2 7 R;U 9�9 .)In fact, in the two theoremsabove, alsoseveralothersubclassesof MTI can

take the placeof ª (seeTheorems5.2.8and5.2.7). We alsostatea few otherresultsconcerningthespectralfactorizationof TI mapsandsomeresultsonothersubclasses.

If theassumption “ &µ4¯ª7 U 9 ” is replacedby “ &µ4 TI 7 U 9 ”, thenthe “gener-alizedcanonicalfactors” , and ( of & needno longerbestablein the indefinite


case(but their Cayley transformsare invertible in H2 over the unit disc). FordimU [ ∞, thiscanbefoundin [CG81]or in [LS] (with theCayley transformsof#, Î 1 and #( Î 1 beinginvertible in H2 over theunit disc). We show thatthis theoryhasanextension for thecasewhereU is anarbitraryHilbert space(seep. 148andTheorem9.14.6).

To emphasizetheimportanceof spectralfactorization,wenotethatoneof themainthemesof thismonographis theequivalenceof thefollowingfour conditionsfor severalcontrolproblemsfor anexponentially stableWPLS:

(I) theproblemhasa (nonsingular)solution;

(II) thePopov Toeplitzoperatorof theproblemis invertible;

(III) thePopov operatorof theproblemhasaspectralfactorization;

(IV) theRiccatiequationof theproblemhasastabilizing solution.

For thecasewheretheWPLSis merelystable,wegetalmostthesameresultsand the unstablecaseis somewhat analogous(it can often be reducedto the[exponentially] stablecase).

For systems with a I/O mapin MTIC (andhencethePopov operatorin MTI),theequivalence“(II) Ñ (II I)” follows from eitherof the two theoremsabove (theformeronecoversmoreclassesof I/O mapsbut is only applicablein minimizationproblems).

Theequivalence“(I) Ñ (II) ” will beestablishedin Chapter8 andin thesectionscorrespondingto the particularcontrol problems; equivalence“(III) Ñ (IV)” willbe establishedin Section9.1 (assuming sufficient regularity of the I/O mapandthespectralfactor;MTI mapsaresufficiently regularfor ourpurposes;hence,forsuchsystems,wehaveacompleteequivalenceof (I)–(IV)).

The I/O map of a finite-dimensional systemis rational, hencein MTI (ifstable).Therefore,in thestandardfinite-dimensional theorywe alwayshave theequivalenceof (I)–(IV).

Theorem1.2.5 is not true for ª½$ TI, not even whenU $ C2 (by Exam-ple 8.4.13),andthe equivalence“(III) Ñ (IV)” doesnot even hold for all regularsystems(by Proposition 9.13.1(c1)). For thesereasons,someof our resultsinChapters9–12poseadditionalregularityassumptionsonthesystem; mostof themaresatisfiedby systemshaving aMTIC I/O map(cf. Theorem8.4.9).

Part II: Continuous-Time Control Theory

This part containsthe theory of well-posedlinear systems(WPLSs): systemtheory, regularity, spectralandcoprimefactorizationandstabilization(by staticfeedback,statefeedback,outputinjectionor dynamic feedback).

Chapter 6: Well-PosedLinear Systems(WPLS)

Chapters6 and 7 presentan extensive theory on Well-PosedLinear Systems(WPLSs): state-spaceandfrequency-domaintheory, stability, regularity, factor-


ization,statefeedback,outputinjection,staticanddynamicoutputfeedbackandrelationsto Pritchard–Salamonsystems andotherspecialcases.

Someof theresultsin thesechaptersareratherstraight-forwardextensionsofexistingtheoryor generalizationsof classicalresults,thoughyetusefulfor controlproblems.Themainnew contributionsof Chapter6 includethefollowing (in theorderof appearance):

1. the relationsbetweenthe stabilities of different partsof a WPLS (fromLemma6.1.10to Example6.1.14);

2. several,oftenverytechnicalregularityresultsneededin theRiccatiequationtheory;

3. compatibility theory(to write alsoirregularWPLSsin adifferentialform asin (1.7));

4. infinite-dimensional quasi- and pseudo-coprimefactorizationtheory andcorrespondingstabilization theory (Sections6.4–6.7). This theoryservesalmostaswell as the classicalcoprimefactorizationtheory for the stabi-lizability anduniquenessanalysisof thesolutionsof Riccatiequations,butthesestrictly weaker coprimesspropertiesaresometimesmoreeasilyveri-fied,andquasi-coprimenessis preservedunderdiscretizationin bothdirec-tions,thusallowingoneto reduceseveralproofsto discretetime.

5. new resultson the generatorsof closed-loopsystems(part of Proposition6.6.18);

6. equivalent conditionsfor different stability and stabilizability properties(particularlypartsof Theorems6.7.10and6.7.15);

7. theory of systemswith a smoothing semigroup(Section6.8, particularlyLemma6.8.5);

8. thecharacterizationof thosetransferfunctions(equivalently, of I/O maps)that have realizationshaving a boundedinput or output operatoror aPritchard–Salamonrealization(Theorems6.9.1and6.9.6);

Alsoalmostall of ourresultsin Chapters6–12will begivenin aWPLSsetting,thereforewemotivatethesesystemsbriefly below.

Lineartime-invariantcontrolsystemsareusuallygovernedby theequations

xn 7 t 9¢$ Ax7 t 9 À Bu7 t 9�1 y 7 t 9$ Cx À Du 1 x 7 09¢$ x0 7 t b 09�1 (1.7)wherethe generators i A BC D l 47 H y U 1 H y Y 9 of the systemarematrices,ormore generally, linear operatorsin Hilbert spacesof arbitrary dimensions, andu : R V U is theinput,x : R V H is thestateandy : R V Y is theoutputofthesystem.If thegeneratorsarebounded,thenthesolution of (1.7) is obviously


B Ò τÓ .·x0 ·u¸ x x $)B x0 À Ò τu¸ y y $ Ó x0 À . u

Figure1.1: Input/state/outputdiagramof a WPLS Fut vcwI Jgivenby thesystemÔ

x 7 t 9Õ$ B37 t 9 x0 À Ò τ 7 t 9 uy $ Ó x0 À . u 1 where (1.8)B37 t 9$ eAt 1 Ò τ 7 t 9 u $? t

0B37 t E s9 Bu7 s9 ds1Ó

x0 $ CB37�`a9 x0 1 . u $ CÒ τ 7 t 9 u À Du © (1.9)The formulae(1.8)–(1.9)areactuallyvalid for ratherunboundedgenerators.

Therefore, WPLSs are defined by requiring B to be a strongly continuoussemigroup,. to be time-invariant and causal,Ò and Ó to be compatiblewithB and . , and F t § t ¨ w τ § t ¨v I J beinglinear andcontinuousH y L2loc 7 R ;U 9 V H yL2loc 7 R ;Y 9 for eacht b 0, equivalently, that] x 7 t 9O] 2H À t

0] y 7 s9] 2Y ds Ö Kt �] x0 ] 2H À t

0] u 7 s9] 2U ds (1.10)

for some(equivalently, all) t 0, whereKt dependson t only. An equivalentformulationis given in Definition 6.1.1,wherewe usethe uniquenaturalexten-sionsof Ò and . thatallow the inputsto be definedon thewhole real line, thussimplifying severalformulae.

Abstract linear systemtheory has beengradually developed since RudolfKalman’s work in [KFA], by William Helton [Helton76a],Paul Fuhrmannandothersuntil DietmarSalamonandAnthony Pritchard[PS85] [PS87] formulatedthe Pritchard–Salamonsystems, which are formally close to WPLSs. Thesesystemshave beenextensively studiedin eightiesandearlynineties,but they donot cover all interestingexamples.This motivatedSalamonto defineWPLSsin[Sal87].

TheLax–Phillipsscatteringtheory[LP] andtheoperator-basedmodeltheoryof BélaSz.-NagyandCiprianFoiaş [SF] gavearemarkableimpactto theresearchalreadyon theseventies,andthesetheorieshave beenshown to beequivalent toWPLSs(seeChapter11 of [Sbook]). Thus,alsothesystemtheorybasedon theLax–Phillipsmodelandextensively developedin Soviet Unionby D.Z. Arov andothers(independentlyfrom WPLSs;see[AN] andits referencelist) hasexactlytheWPLSframework.

Until then,researchhadbeendividedby differentwaysto representasystem,for example:

(1.) in termsof partial differential equationsor differential delay equations[Lions] [FLT],

(2.) in termsof thegeneratorsi A BC D l [Helton76a][Fuhrmann81],


(3.) asa frequency domainrelationshipbetweeninputs andoutputs [CG97],

(4.) asadynamicalsystem(e.g.,WPLS)in thesenseof Kalman[KFA],

(5.) by fractionalrepresentations[Vid] [CD78]

as notedby Ruth Curtain [Curtain97], who emphasizedthe needfor a theorycoveringbothstate-spaceandfrequency-domainaspectandunifying all theaboverepresentations;the work of Salamonand George Weiss in the late eightiesshowed that WPLSs satisfy this need. ThereafterWPLSs have becomeanincreasinglypopular subject in someparts of control theory, being the mostgeneralwidely-usedclassof infinite-dimensionallinearsystems.

The more specializedapproachesstill have their advantagesin the studyof special cases. One of the most important examples of this is the workof Irena Lasiecka, Roberto Triggiani and others (see [LT00a], [LT00b] andreferencestherein),who have solved statefeedbackproblemscorrespondingtoseveral importantPDEsandrathercoercive cost functions,by usinga moreadhoc approach(of type “(1.)”). At its best, the abstractWPLS approachcancomplementthe othersby providing a different insight and an abundanceofresultsincluding thosecommonfor rathergeneralsystemsand cost functions,thusremoving theneedto “reinventthewheel”over andoveragain.

We studythe basicproperties,stability, realizationtheory, dual systemsandgeneratorsof WPLSsin Section6.1. For any WPLS, therearegeneratorsB 47 U 1 Dom7 A *�9�*�9 andC 4¯7 Dom7 A9�1 Y 9 satisfying (1.9) in a strongsense(e.g.,\ t0 B37 s9 Bu7}E s9 ds converges in Dom7 A *�9�* but its value belongsto the smallerspaceH andequalsÒ u; alsothe formulaxn $ Ax À Bu holdsin Dom7 A * 9 * a.e.),asshown by Salamon[Sal89]andWeiss[W89a] [W89b]. Salamonalsoobservedthatany TIC∞ map(or propertransferfunction)canberealizedasaWPLS.

A WPLS neednot have a well-definedfeedthroughoperator(“D”), but allsystemsof practicalinterestseemto have one; suchWPLSsarecalled regular.Regularity is treatedin Sections6.2 and6.3. An equivalentdefinitionof [weak]regularity is thatthetransferfunctionhasa[weak] limit (necessarilythesameD 47 U 1 Y 9 ) at infinity alongthepositiverealaxis.All weaklyregularsystemssatisfy(1.9) in aweaksense,andtheclassicalformulaesuchas #.87 s9$ D À C 7 s E A9K> 1Bhold if we replaceC by its weakWeissextensionCw.

Regularity is anextremelyimportantproperty, becausefeedthroughoperatorsare of fundamentalimportancefor much of the control theory. For example,optimal control problemsare most often solved throughRiccati equationsthatarewritten in termsof thegeneratorsof thesystem,including the (feedthrough)operatorD.

For generalWPLSs,equations(1.9)andtheclassicalformulaesuchas #.87 s9!$DÀ

Cext 7 s E A9K> 1B still hold in a very weaksensefor certaincompatible pairs7 Cext 1 D 9 ; their theoryis developedin Section6.3,which alsocontainsadditionalresultson differentformsof regularity, on Hp transferfunctions,on therelationsbetweenaWPLSandits generatorsandonreachabilityandobservability.

In Sections6.4 and6.5, we defineandstudycoprime,spectraland losslessfactorizations.The importanceof thesefactorizationsis dueto the equivalenceon p. 21, with coprimefactorizationtaking the placeof spectralfactorizationin

1.2. A SUMMARY OF THIS BOOK 25.Q ×Ø·xy ¸ yÁ

yL

Áu ×ØÄ xu¸ uL

Figure1.2: Dynamicoutputfeedbackcontroller Q for .54 TIC∞ 7 U 1 Y 9“(III)” in the unstablecase,anddue to the strongconnectionbetweencoprimefactorizationand dynamic stabilization. We also presenttwo weak forms ofcoprimeness,which are useful in the infinite-dimensional settings, the weakerof thembeing invariantunder(inverse)discretizationandhenceallowing us toreduceseveralresultsto thesimpler discrete-time theory.

Thus,theconnectionbetweenpresentations(2.)–(5.)of p. 23 is establishedinSections6.1–6.5.Connectionto (1.) is beyond the scopeof this book. Instead,westudyWPLStheory, with emphasisonRiccatiequationsandoptimal control.

Sections6.6 and6.7 treat statefeedback,output injection and staticoutputfeedback. Sinceour interestis not limited to exponentialstabilization, but weoften only require that the controller makes the closed-loopsystemstableorstrongly stable(this hasbecomeincreasinglypopular lately), we meetcertainadditionaldifficulties.

In Section6.8,westudysystemswhosesemigroupis smoothing (e.g.,B Bu0 4H a.e.on R for eachu0 4 U ). In Section6.9,we show thata transferfunction#. hasa realizationwith boundedB if f #.ÐE #.87 À ∞ 9=4 H2strongoversomeright half-plane.Wealsoestablishanalogousresultsfor realizationswith boundedC andforPritchard–Salamonrealizations.

Chapter 7: Dynamic Stabilization

In this chapter, we treat different forms of dynamicstabilization. In dynamicoutput feedback (Section7.1), the output is fed back to the input through aDynamicFeedback Controller, in order to stabilize andcontrol the plant, as inFigure1.2.

As one can verify from Figure 1.2, the map from the original input to theoutputof theplant . : u TV y becomes.37 I E¤Q�.{9Z> 1 : uL TV y.

We have above treatedonly the I/O mapsof the plant andof the controller.We shall also study the problemwherethe plant and the controllerhave to bestabilizedinternally too (seeFigure 7.2), but mostsuchresultsare obtainedascorollariesof theI/O theory, sinceacontrollerstabilizesasystemexponentially if fit I/O-stabilizesthesystemandboththesystemandthecontrollerareoptimizableand estimatable (this is an extensionof the classicalconcept“exponentiallystabilizableand exponentially detectable”),as shown in [WR00], cf. Theorem7.2.3(c1).

The main new contributions of this section include the relationsbetweenexternalandinternalstability of thecontrolledsystem(Theorems7.2.3and7.2.4),particularlytheextensionof theequivalence(1.11);certainresultsof theinternalloop theoryrequiredby theH∞ 4BPtheory, includingthecorollarieson dynamicpartial feedback;andtherelationbetweenthestabilization of thecontrolledpart


and the stabilization of the whole plant in partial feedback(Lemmas7.3.5and7.3.6andTheorem7.3.11).

In Chapter7, we extend most classicalresults(such as the connectiontocoprimefactorizationandYoulaparametrizationof all stabilizing controllers)tothe infinite-dimensional caseand presentsomenew results. For example, weextend(seeTheorem7.2.4(c))theclassicalequivalence

exponentially DF-stabilizable ÙÛÚ exponentially stabilizableanddetectable(1.11)

to a largesubclassonWPLS(includingtheparabolicsystemsof Section9.5).In Section7.2,westudythemoregeneralcontrollerswith internalloop, whereQ neednot be well-posed(i.e., proper), as long as the closed-loopsystemis

still well-posed;classical“fractional H∞ Ü H∞” controllersfall into this category.For example, if .4 TIC∞ 7 U 1 Y 9 has the doubly coprime factorization (d.c.f.).:$0;= 1 $ x 1 x; , where


(or makesit lessthanagivenconstantγ 0).Part III: Riccati equations and Optimal control

Thispartcontainsatheoryonoptimalcontrol(bothin anabstractsetting,andasanapplicationto WPLSs)andRiccati equations,with applicationsto minimization(LQR andH2) problemsandto theH∞ full-informationandfour-blockproblems.

Chapter 8: Optimal Control ( ddu æÌç 0)We presentan abstracttheory on optimization and optimal control in statefeedbackform (Sections8.1and8.2)andtheapplicationof this theoryto WPLSs(Sections8.3and8.4)with guidelinestoproblemsfinite timeinterval (Section8.5)andto systemswheretheinputoperator(B) is allowedto bemoreunboundedthanthat of WPLSs(Section8.6). We solve the generalizedcontrol problem,whose(possiblyindefinite)costfunctioncoversmoststandardcontrolproblems.

Our main contributionsincludethe generalizationof the classicalcoercivityassumptionto generalWPLSsandcostfunctions,andthefactthatthisassumptionleads to a solution of the generalizedcontrol problem (see Theorems8.4.3and 8.3.9); this was alreadyextendedto stableWPLSs by O. Staffans. Animportantpart of our theory are also the methodsto treatesimultaneouslyallformsof stabilization(i.e.,whetheronerequiresthe“optimal control” to be,e.g.,exponentially, strongly or merelyoutput-stabilizing). Theseresultswill thenbeappliedin the derivation of the Riccati equation,LQR and H∞ theoriesin thechaptersto follow.

We studythecritical pointsof a givencostfunctionandthecasewheresuchcontrol correspondsto a stabilizingstatefeedbackpair. Suchan “optimal” statefeedbackpair correspondsto a “stabilizing” solutionof the Riccati equation,asshown in Chapter9. The correspondingspecialcontrol problemsaresolved inChapters10–12.

GivenaWPLS èpé êë INì andacostoperatorJ í J î=ïÀðñ Y ò , weconsiderthecostfunctionó ñ x0 ô uò : í5õ ∞

0 ö y ñ t ò ô Jy ñ t òP÷ Y dt ô where y : í:ø x0 ùú u ñ x0 ï H ô u : R û¯ü U ò(1.15)

andu is requiredto be exponentially stabilizing, stronglystabilizing, stabilizingor something similar, dependingonhow stableonewishestheclosed-loopsystemto be.

This coversall quadratic(definiteor indefinite)costson the input, stateandoutput(extend ø and ú suitably if necessary, e.g., replaceø by è ë0 ì and ú byè I I ì to cover crosstermsof u andy). In particular, minimization,H∞ andsimilarcontrolproblemsarecovered.Thesolutionsof suchproblemscorrespondto thecontrolsthatarecriticalpointsof

ó, i.e.,for whichtheFréchetderivativeof

ó ñ x0 ô�ý òis zero;wecall suchcontrolsJ-critical.


In Section8.4,we defineandstudyJ-coercivity, which is a generalizationofthe standardnonsingularity assumptions of several control problems(includingthe “J-coercivity” assumptions definedin [S97b]–[S98d],the “Popov Toeplitzinvertibility” conditionin thestablecaseandthe“no transmission zeros”and“noinvariantzeros”conditionsin thepositive case).We show thatany “stabilizable”J-coerciveWPLShasauniqueJ-critical (“optimal”) controlfor eachinitial state,andthat this J-critical control canbe presentedin WPLS form (this generalizesthecorrespondingresultin [FLT]).

However, the correspondingfeedbackneednot be well-posedwithout addi-tional assumptions on the system,as illustratedin Examples8.4.13and11.3.7.This leadsto someadditionaldifficulties in the Riccati equationtheory(the sit-uationis the sameeven in the casestudiedin [FLT]). Sections8.3 and8.4 alsocontainsaseriesa furtherresultsonJ-critical controlsandJ-coercivity andontheconnectionof thelatterto spectralandcoprimefactorizations.

The control problemsfor unstable systemsare traditionally reducedto thestablecaseby preliminarystabilization, whenthe optimal control is requiredtobe exponentially stabilizing. We show that this is possible for WPLSstoo, givea counter-examplefor otherformsof stabilizationanddevelop morecomplicatedtricks to overcomethisproblem(Theorem8.4.5).

In the last two sectionof Chapter8, we give guidelines on how to extendour optimizationandRiccatiequationresultsfor problemson finite time interval(Section8.5) and for more generalsystemsthan WPLSs(Section8.6). Theseresultsarenotusedelsewherein thismonograph.

Chapter 9: Riccati Equations and J-Critical Control

It was shown independentlyin [WW] and [S97b]–[S98d]that, in the (stable)regular case,the optimal cost operatorof certain control problemssatisfiesageneralized(operator)Riccatiequation.We establishedtheconverseimplicationfrom a stabilizingsolution of theRiccati equationto theexistenceof anoptimalcontrol in [Mik97b]. In Chapter9, we extend both results to the generaloptimization context of Chapter8, thuscovering alsogeneralunstablesystemsandmoresingular problems(underweaker regularityassumptions).

We alsosimplify the equationand the assumptions in several specialcases,presenta priori sufficient assumptions for the requiredregularity, and provideweaker resultsfor lessregular settings.Moreover, the connectionto spectralorcoprimefactorizationandfurther aspects(suchasuniqueness,Riccati inequali-ties andcertainpathologies) areaddressed.Possiblyill-posedor irregular opti-malcontrolsandcorrespondinggeneralizedRiccatiequationsarecoveredin Sec-tion 9.7(for boundedoutputoperators,aspecialcaseof thiswassolvedin [FLT]).We describebelow themainresultsof thischapter.

Theexistenceof auniqueregularoptimalstatefeedbackoperatorfor aregularWPLS is equivalent to the existenceof a (necessarilyunique) þ¶îî -stabilizingsolutionof theContinuous-timeAlgebraic RiccatiEquation (CARE)andfrom onethe othercanbe computed(seeTheorem9.9.1; read“optimal” as “J-critical”).This extendsmostsimilar resultsin theliterature.


Whenwe optimize over exponentially stabilizing controlsor statefeedbackoperators,theterm“ þ¤îî -stabilizing” is equivalentto “exponentially stabilizing” (aWPLSisexponentially stableif f itssemigroupÿ satisfies�Kÿ3ñ t ò�� H � Meωt ñ t � 0òfor someω � 0, M � ∞). To make thingseasier, we illustratethis underratherstrongassumptions:

Theorem 1.2.6( þ expþ expþ exp: Unique minimum � BîwBîwBîw-CARE � J-coercive) Assumethat the WPLS èé êë INì and the cost operator J í J î'ïÌðñ Y ò are s.t.π � 0 � 1 � ÿ B ï L1 ñ 0 ô 1ò ; ð¯ñ U ô H ò�ò , C ïÛðñ H ô Y ò andD î JD � 0. Thenthefollowingareequivalent:

(i) There is a uniqueminimizing exponentially stabilizing statefeedback opera-tor.

(ii) There is a uniqueminimizing control over þ exp ñ x0 ò : í� u ï L2 ñ R û ;U ò�� x ïL2 ñ R û ;H ò�� for each initial statex0 ï H.

(iii) TheRiccatiequationñ Bîw � ù D î JCò î ñ D î JD ò�� 1 ñ Bîw � ù D î JCò�í Aî � ù � A ù C î JC (1.16)hasa solution � í � î ï±ðñ H ò s.t. � H �� Domñ B îw ò andthesemigroupgen-eratedby A E BK is exponentiallystable, where K : í¥EÁñ D î JD ò � 1 ñ Bîw � ùD î JCò .

(iv) Σ is optimizableand ú is J-coerciveover þ exp.(v) Σ is exponentiallystabilizableandthere is ε � 0 satisfyingñ ir E Aò x0 í Bu0 � öCx0 ù Du ô J ñ Cx0 ù Duòp÷�� ε � x0 � 2H ñ x0 ï H ô u0 ï U ô r ï R ò��If (iii) holds,thenK is bounded(K ïÁðñ H ô U ò ) andit is theuniqueminimizing

exponentially stabilizing state feedback operator. The minimal cost equalsö x0 ô � x0÷ for each x0 ï H. �(This is aspecialcaseof Corollary10.2.9combinedwith Theorem9.2.3.)Thus,theoptimalcontrolcorrespondsto thestatefeedbacku ñ t ò¢í Kx ñ t òÁñ t �

0ò , where K is as above. Here Bîw denotesthe Weiss extension of B î ïðñ Domñ A î�ò ô U ò . The Riccati equation(1.16) is given on Domñ Aò (see(9.14)).See(1.17)for themorecomplicatedgeneralCARE.

When

ó ñ x0 ô uòí�� Cx � 22 ù � u � 22, i.e., C í¿è C10 ì , D í¿è 0I ì , J í I , then(1.16)becomesñ Bîw � ò�î Bîw � í Aî � ù � A ù C î C, the minimizing feedbackis givenby u ñ t ò�í½E B îw � x ñ t ò¤ñ t � 0), and the closed-loopsemigroupis generatedbyA ù BK í A E BBîw � .

As explainedon p. 27, we canhave crosstermsof u andy in the cost,e.g.,replaceC by è C0 ì andD by è DI ì to obtainanotherWPLSand,correspondingly, a“moregeneral”(actually, lessgeneral)“standard”form of theRiccatiequation,asin, e.g.,Remark9.1.14.

However, thetheoryof Section8.3alsoallowsoptimizationovervariousothersets(“ þ¯îî ”) of controlsthan þ exp, e.g.,for thosewhichmake thestateandoutputstronglystablefor eachinitial state(“ þ str”). Correspondingly, theregularoptimal


statefeedbackoperator(if any) over þ str correspondsto theuniquesolutionof theCARE thatis þ str-stabilizing, i.e., thatstabilizesthestateandoutput stronglyforeachinitial state.

In the literatureof infinite-dimensional systems,it hasbecomepopular toonly requirethat the output is stablefor eachinitial stateandpossibly also foreachstableexternalinput to the feedbackloop. In this casethe condition “ þ·îî -stabilizing”becomesrathercomplicated(Definition9.8.1).

If thesystemis exponentially detectable,thenall thecasesmentionedabove(andcertainothers)coincidewith exponentialstabilization,but this assumptionis sometimestoo strong. If the systemis “coprime stabilizable” (in a suitable,rather weak sense;this assumptionalways holds when the systemis outputstable(resp.stable,stronglystable)),thenoptimization over output-stabilizable(resp.stabilizable, strongly stabilizable) controls correspondsto the “coprimestabilizing”solutionof theCARE,andtheequivalenceof (I)–(IV) onp. 21holds,seeSection9.1 for details. However, this solution neednot be exponentiallystabilizing, and the sameCARE may also have an exponentially stabilizingsolution(seeExample9.13.2;naturally, in a minimization problemthe optimalcostbecomeshigherfor strongerstabilizability requirements).Partof theseresultsseemto benew evenfor finite-dimensionalsystems.

Very regular systems, suchas thoseof Theorem1.2.6, are studiedin Sec-tion 9.2. For themthe CARE becomesratherelegantand similar to its finite-dimensionalcounterparts,aspart(iii) of thetheoremshows. Suchsystemscoveranalyticsystems(hencemostparabolic-typeproblems)having ratherunboundedinputandoutputoperators,asshown in Section9.5.

In thegeneralcase,theoptimalcontrolneednot correspondto a (well-posed)statefeedbackoperator, as explainedin Chapter8. Nevertheless,suchcontrolcorrespondsto a generalizedRiccati equation,as illustratedin Section9.7 (forWPLSswith aboundedoutputoperator(“C”) anda rathercoercivecostfunction,this wasshown in [FLT] by F. Flandoli, I. LasieckaandR. Triggiani). However,since theseequationsare given on the (unknown) domain of the closed-loopsemigroupgeneratorratherthanonDomñ Aò , it becomesverydifficult to solvetheRiccatiequationandthusobtainthe(possibly non-well-posed)feedbackoperator.

As mentioned above, the existenceof a (well-posed)regular statefeedbackoperatorfor a regularWPLSis equivalent to theCARE having a solution, but inthis generalcasetheCARE becomesrathercomplex: we have to find � í � î/ïðñ H ò satisfying� ! " K î SK í Aî � ù � A ù C î JC ïðñ Domñ Aò ô Domñ Aò î òS í D î JD ù w-lims# û ∞ Bîw � ñ s E Aò � 1B ïðñ U ò

SK íEÁñ Bîw � ù D î JCò ïðñ Domñ Aò ô U ò�� (1.17)Obviously, S and K are uniquely determinedby � if S is one-to-one,whichcorrespondsto a uniqueoptimal control. The optimal statefeedbackis givenby u ñ t òcí Kwx ñ t ò for a.e.t � 0. SeeDefinition 9.1.5for details(andDefinition9.8.1for noninvertiblesignatureoperators).

Note that whereasthe special case(1.16) is close the finite-dimensional


CARE, this generalform looksalmostlike thediscrete-timeRiccatiequation;inparticular, thesignatureoperator Smaydiffer from D î JD, asobservedin [S97b]and[WW]. In the notesto Section9.8 we explain how the signaturepropertiesof the problemaredeterminedby S, not by D î JD, even whenthe I/O mapis asimpledelay. Thus,thesituation is analogousto the(finite-dimensional)discrete-time setting,wherethe signature operatorS : í D î JD ù Bî � B takes the role ofD î JD.

We alsolist severalcasesin which theCARE canbesimplified andcasesinwhich anoptimalcontrol is alwaysgivenby a well-posedregularstatefeedbackpair (andhencecorrespondsto aCARE; see,e.g.,Remark9.9.14).

The optimal control is given by a well-posedstatefeedbackif f the IntegralAlgebraic RiccatiEquation(IARE) hasan þ îî -stabilizing solution,regardlessofregularity. While IAREs arenot particularlyapt for engineeringpurposes,theyprovide a link to discrete-timeRiccati equations,and this allows us the proveseveral resultswhosecontinuous-timeproofswould seemintractabledueto theunboundednessof input andoutputoperators.The IAREs alsoallow us to treatthe connectionbetweenoptimal control and Riccati equationsseparatelyfromregularity considerations. Naturally, for regular WPLSs, the solutionsof theCARE areexactly thesolutionsof the IARE correspondingto regular feedback.Also thesequestions are addressedin Section9.8. Several further propertiesof Riccati equationsare treatedin the restof the chapter. Much of our theoryconcerningfor þ îî%$íþ exp is new evenfor finite-dimensional systems.

In Section9.14,we give anextension of thegeneralizedcanonicalfactoriza-tion theoryto the caseof infinite-dimensional input andoutputspaces(seealsop. 148).

Chapter 10: Quadratic Minimization (LQR)

For controlproblemswith a positivePopov operator, onetraditionally shows thatundercertainconditionsany solution of theRiccatiequationis unique,admissibleandexponentially stabilizing. Oneof ourmaincontributionsin thisandprecedingchapteris the extension of the above fact to WPLSsand partially also to thenon-exponentially stabilizing case;this is technicallyvery challengingdueto theunboundedinputandoutputoperators,which,e.g.,makeit hardto show whenthe“optimal feedback”is well posed.

As corollaries,we get several resultsthat formally look like the classicalones. ThesecorollariesincludeTheorem1.2.7below, (b4)&(c1)&(c2) of The-orem 10.1.4,the Strict Boundedand Strictly Positive (Real) Lemmas,and theequivalencebetweenoptimizability and exponentialstabilizability for systemswith a smoothing semigroup(Theorem9.2.12).We alsosolve severalminimiza-tion problemswith moregeneralstabilizability or regularityassumptions.

Importantnew contributions of the chapteralso include the connectionbe-tween different classical coercivity assumptions and their generalizationstoWPLSs,includingJ-coercivity (Section10.3).

In Section10.2,we studyminimizationproblems,by which we refer to theminimization of the cost function (1.15). Theorem1.2.6 is a corollary of that


section.In Section10.1westudythespecialcaseof thecostfunction � y � 22 ù � u � 22 and

its variants.Underamild detectabilitycondition,thereis atmostonenonnegativesolutionof theCARE,hencewe do nothave to verify whethera solution is “ þ·îî -stabilizing”:

Theorem1.2.7(LQR: minu & ∞0 ñ�� x � 2H ù � u � 2U òminu & ∞0 ñ�� x � 2H ù � u � 2U òminu & ∞0 ñ�� x � 2H ù � u � 2U ò ) Assumethat the WPLS Σ íèé êë INì is uniformly regular (UR) and estimatable (e.g., that C is boundedandC î C � 0). Consider, for someR ïÛð¯ñ U ò , Q ïÛð¯ñ Y ò s.t.Rô Q � 0, thecostfunc-tionó ñ x0 ô uò : íµõ ∞

0 ' ö y ñ t ò ô Qyñ t òp÷ Y ù ö u ñ t ò ô Ruñ t òp÷ U ( dt ñ x0 ï H ô u ï L2loc ñ R û ;U ò�ò��(1.18)

There is a UR minimizing state feedback operator for Σ iff there is anonnegativesolution � ïoðñ H ò satisfying theCARE� ! " K î SK í Aî � ù � A ù C î QC ôS í R ù D î QD ù lims# û ∞Bîw � ñ s E Aò � 1B ô

K íE S� 1 ñ Bîw � ù D î QC ò ô (1.19)for someK ïoð¯ñ H1 ô U ò , S ïðñ U ò , S � 0.

If such a solution exists,thenit is theuniquenonnegative solution of (1.19),K is a UR exponentiallystabilizing statefeedback operator for Σ, and K is theuniqueminimizing statefeedback operator over all u ï L2loc ñ R û ;U ò (and overþ exp andover þ out). �

(This follows from Theorem10.1.4and Remark10.1.5.) For eachCAREresult in this monograph,including the oneabove, thereis alsoa “B îw-CARE”variantthatallowsusto removethelimit termandsimplify theformulationunderany of the regularity assumptionsof Hypothesis 9.2.2,asillustratedin Theorem1.2.6.

Without the detectability (estimatability) condition,we observe that a mini-mizingstatefeedbackoperatorover þ exp correspondsto themaximalnonnegativesolutionof theCARE anda minimizing statefeedbackoperatorover þ out corre-spondsto theminimal nonnegative solution of theCARE (Theorem10.1.4).Wealsoderive furtherresultsonsuchandmoregeneralminimizationproblems.

In Section10.4we show that thesolution of theminimizationproblemleadsto thesolution of theH2 full informationandstatefeedbackproblems,whereonewishesto find a controller ( ) ; possiblyinducedby statefeedbackor dynamicoutputfeedback)thatminimizesthenorm��*ú *) ù *ú 2 � H2strong+ C , ; - +W�Y �.� ô (1.20)where *) : *w ü *u is the frequency-domaincontrol law (determinedby ) ) to anexternal(disturbance)input to thecontrolinput for theWPLS/ ÿ 0 0 2ø ú ú 2 1 with generators / A B B2C D 0 1 ô (1.21)


asin Figure10.1. Theabove WPLS is obtainedby addinga secondinput to theWPLSΣ, Weassumethatboth ú andú 2 areWR.A strongerproblemis tofind, foreachw0 ï W, a “stabilizing” controlu s.t. � *ú *u ù *ú 2w0 � H2 + C , ;Y � is minimized,seeSection10.4for details.We show thatunderminimal assumptions,a minimizingstatefeedbackoperatorfor theoriginalsystemalsosolvestheH2 problemandthestrongerproblemformulatedabove.

In Section10.3wetreatmoststandardassumptionsfor classicalminimizationproblemsand show that they are stronger than or equivalent to positive J-coercivity (over þ exp or over þ out).

In Section10.5,wepresentgeneralizedversionsof theBoundedRealLemma,includingthefollowing:

Theorem 1.2.8(GeneralizedStrict BoundedRealLemma) Assumethatγ � 0.If C is boundedand dimY � ∞, or if B is bounded,then the following are

equivalent:

(i) Σ is exponentially stableand � ú �2� γ;(ii) There is � � 0 s.t. � H �3� Domñ B îw ò and/

Aî � ù � A E C î C ñ Bîw � E D î C ò�îBîw � E D î C γ2I E D î D 1 � 0 on Domñ Aò54 U � (1.22)

Moreover, anysolution of (ii) satisfies� � 0.In theStrictPositiveRealLemma,wepresentanalogousconditionsfor theI/O

mapto satisfyú ï TIC andReö ú�ýpô�ý ÷�� 0 (i.e., *ú¯ù *ú î6� εI in L∞strongñ iR; ð¯ñ U ô Y ò�òfor someε � 0). Naturally, therearealsoanalogousresultsfor unboundedB andC.

In Section10.6,wepresentnecessaryandsufficientconditionsfor theuniformpositivity of the Popov operator( ú î J ú � 0), in termsof spectralfactorizationsandRiccati equationsor inequalities. Section10.7presentadditional resultsforpositiveRiccatiequations(say, with positivesignatureoperator, S � 0).Chapter 11: The H∞ Full-Inf ormation Control Problem(FICP)

TheH∞ controlproblemsrefer to theminimizationof theoutputof a plantin thepresenceof a disturbanceinput. The name“H ∞” comesfrom the minimizationof the(controlledclosed-loopsystem) L2 ü L2 normfrom disturbanceto output,which equalsthe H∞ norm of the correspondingtransferfunction, by Theorem1.2.1.

In the FICP, studied in this chapter, we can produce the control signalknowing exactly the stateand disturbanceof the system, whereasin the Four-Block Problemof Chapter12thecontrollersonly input is aseparatemeasurementoutput.

Our mainresultsstatethat,givenγ � 0, thereis a controller achieving a normlessthanγ if f the Riccati equation(1.24)hasa nonnegative stabilizingsolution.Moreover, if this is thecase,thenthereis sucha controllerthatconsistsof pure


ÿ 0 1τ 0 2τø ú 1 ú 271 8 1 8 29 x9 y :

; ûû u , with input spaceU 4 W insteadof U . If ú í è ú 1 ú 2 ì ï TIC∞ ñ U 4 W ô Y ò is regular, wecanwritethisas ?

x@äí Ax ù B1u ù B2wôy í Cx ù D1u ù D2wô (1.23)

(if B í è B1 B2 ì ïðñ U 4 Wô H � 1 ò andC ïÛðñ Domñ Aò ô Y ò areunbounded, thenthedynamics(1.23)aresatisfiedonly in thesensedescribedin Theorem6.2.13).

We have dividedthe input spacein two to modela settingwhereonly partoftheinput(calledthecontrol), u : R ûü U , is accessibleby thecontroller, whereastheotherpartrepresentsthedisturbance(or uncertainties,sensornoise,modelingerror) w : R û·ü W to the system. The signaly is the objectiveor error signalwhosenormis to beminimized.

In theoptimal H∞ State-FeedbackControl Problem(SFCP), onewishestofinda(pure)statefeedbackcontrollerof form “u ñ t ò í Kx ñ t ò ” (with e.g.,K ïÁðñ H ô U ò )such that this feedbackstabilizes the systemexponentially and minimizes thenorm � w Aü y � - + L2 � L2 � . In the optimal H∞ Full-Information Control Problem(FICP), the controller is allowed to be of form “u ñ t ò»í K ñ t ò x ù F2w ñ t ò ” (statefeedbackplus feedforward),asin Figure1.3. (Here è 7 8 1 8 2 ì is the statefeedbackpair generatedby K or è K 0 F2 ì , andthesignalu < representstheexternaldisturbances(or externalinputs)in the feedbackloop. The words“fullinformation” refer to the fact that the controllerhasaccessto both the stateandthedisturbance.)

Thereisnodirectmethodavailable(evenin thefinite-dimensionalcase)tofindtheexactoptimum. Therefore,insteadof theoptimal problem,thecorrespondingsuboptimalH∞ problem is usually treatedin the literature. In the suboptimalH∞ problem, we searchfor an exponentially stabilizing controller such that� w Aü y � - + L2 � L2 � � γ, whereγ � 0 is agivenconstant;suchacontrolleris called(γ-)suboptimal. Weextendtheclassicalresultsby showingthatthereis asuboptimalstatefeedbackcontrollerif f theRiccatiequationcondition(iii) below is satisfied.By varying γ we canthenfind an estimateof the infimal γ anda corresponding


(almostoptimal) controller (e.g.,by abinarysearchover γ’s).As mentionedabove, under standardcoercivity assumptions and certain

regularityandnormalization conditions(see,e.g.,Theorem11.1.4),thefollowingareequivalent:

(i) there is a suboptimal control law w Aü u, and ñ A ô B1 ò is exponentiallystabilizable;

(ii) there is a suboptimal state-feedback(plus feedforward) controller u íKwx ù F2x;

(ii’) thereis suboptimal purestate-feedbackcontrolleru í Kwx;(iii) theRiccatiequation� ñ B1Bî1 E γ � 2B2Bî2 ò � í Aî � ù � A ù C î C ô (1.24)

(on Domñ Aò ) hasa nonnegative solution � ï¯ðñ H ò suchthatA EUñ B1Bî1 Eγ � 2B2Bî2 ò � generatesanexponentially stableC0-semigroup.

Moreover, if (iii) holds, then K : íE B î1 � ï ð¯ñ H ô U ò determinesa suboptimal(pure)state-feedbackcontrollerfor Σ (throughu ñ t ò : í Kx ñ t ò/ñ t � 0ò ). A solution� of (iii) is unique.

Herewe have assumedthatB is bounded,D2 í 0 andD î1 èC D1 ì í è 0 I ì ;see,e.g., (11.24)and (11.17) for the unsimplified forms of (iii) and K. (Alsowithout theabove simplifying assumptions,thesuboptimal statefeedbackopera-tor K is exponentially stabilizing (anduniformly regular, thoughnot necessarilybounded),but wemustaddasignaturecondition to (iii); moreover, condition(ii’)becomesstrictly strongerthanthe otherconditions(which remainequivalenttoeachother)unlessa strongersignatureconditionis satisfied.)

We presentanalogousresults under different regularity assumptions, andvariantsfor þ out, þ sta and þ str, i.e., wherethe suboptimal controller needstobe,e.g.,merelystronglystabilizing insteadof exponentially stabilizing. We alsoestablishthe sufficiency of the Riccati equationcondition for arbitrary regularWPLSs(seeLemma11.2.13).In Example11.3.7(c),we show that,however, thisconditionis notnecessaryfor generalregularWPLSs.

In (i), we haveallowedfor anarbitrarycontrollaw L2 ñ R û ;W òBAü L2 ñ R û ;U ò .If sucha control law ) : w Aü u hasa transferfunction(e.g., )Ìï TIC∞ ñWô U ò ),thenthenorm � w Aü y � equals� ú 1 ) ùÊú 2 � TIC +W �Y � , or ��*ú 1 *) ù *ú 2 � H∞ + C , ; - +W�Y �.� .By theaboveequivalence,thisproblem,theFICPandtheSFCPareall equivalent(undersimplifying assumptions and suitable regularity). Thus, if there is anysuboptimal control law (and ñ A ô B1 ò is exponentiallystabilizable),then thereisactuallyacausal,linear, stable,time-invariantcontrollaw thatcanbeimplementedas an exponentially stabilizing state feedbackcontroller (so that )_íÕñ I E8 1 ò � 1 8 2). Condition (i) can also be formulated as a minimax problem, asexplainedin Section11.1(particularlyonpp.613and626).

In Section11.2,we give proofsandadditional variantsfor theabove results,andwe extendthe(frequency-domain)J-losslessfactorizationresultsfor theH∞

FICPgivenin [Green]and[CG97]to MTIC andsimilarclasses(Theorem11.2.7).


TheDiscrete-TimeH∞ FICP is treatedin Section11.5,andtheabstractH∞ FICPin Section11.7.

TheH∞ FICPis interesting bothfor its own meritsandfor thefact that it canbeusedto obtainasolutionto theH∞ 4BPpresentedbelow.

Themethodsusedfor thestableH∞ FICPalsoapplyto the(one-block)Nehariproblem,whereonewishesto estimated ñ ú�ô TIC î ò or theHankel norm � π û ú π � �of someú ï TIC. Therefore,we takeabrief look at thisproblemin Section11.8,this includesthefollowing:

Theorem1.2.9(Nehari) Let ú ï MTIC ñW ô U ò andγ � 0. If dimU 4 W � ∞ orú ï MTICTZ , thenthefollowingareequivalent:(i) There is C5ï TIC ñ U ô W ò s.t. � úÛù C/îD� - + L2 � � γ (i.e., d ñ ú�ô TIC î òE� γ).(ii) TheHankel norm � π û ú π � � of ú is lessthanγ.(iii) There is F ïHG TIC ñ U 4 W ò s.t. F 11 ïHG TIC ñ U ò andè I I0 I ì î F I 00 � γ2I > è I I0 I ì íIF è I 00 � I ì F+î .(Recallthat ú ï MTICTZ meansthat ú hasanL1 impulseresponseplusdelays

of form ∑∞k J 0Dkτ � kT for someperiodT � 0.)Thefactorizationin (iii) is oftencalleda co-spectralfactorization.Thenorm� π û ú π � � equalsρ ñK0L0�î�ø»îø»ò 1M 2, where 0B0+î and ø»î�ø are the reachabilityand

observability Gramians,respectively, of any realizationof ú having stableinputandoutputmaps.

Wedonot treattheNehariRiccatiequations,sincetheir theorywould requirelengthy additions to Chapter9 due to the noncausalityof the corresponding“closed-loopsystems”.

Chapter 12: H∞ Four-Block Problem( N�OQP�RTSVU�WVX5NZY γ)In the H∞ Four-Block Problem(H∞ 4BP) (aka. “the standardH∞ problem” or“the generalregulatorproblem”),onetriesto find aDPF-controllerthatmakesthenormw Aü z lessthana givenconstantγ � 0 (see(1.14)),i.e.,γ-suboptimal.

Consequently, asexplainedabove, thedifferenceto theH∞ FICP is thatnowthecontrollerdoesnothaveaccessto thedisturbance,only to apartof theoutput(“the measurement”),asin Figure1.4(or in Figure7.8;seeFigures7.10and7.11for DPF-controllerswith internalloop).

Thus,thegoalof theengineeris againto minimizethenormfrom theexternaldisturbanceinput to theobjective outputof thesystem. As in thepreviouschap-ter, we againgeneralizetheclassicalresult(previously generalizedto Pritchard–Salamonsystemsby B. vanKeulen[Keu]) that thereis a γ-suboptimal exponen-tially stabilizing (measurementfeedback)controller if f certaintwo independentRiccati equationshave exponentially stabilizing nonnegative solutionsandthese(necessarilyunique)solutionssatisfythe standardspectralradiuscondition. Weformulatetheresultalsoin termsof two nestedJ-losslessfactorizationsandsolvethe H∞ discrete-timeFour-Block Problem;in fact thesetwo generalizationsofclassicalresultsserveaspartsof our lengthyproof.

1.2. A SUMMARY OF THIS BOOK 37ÿ 0 1τ 0 2τø 1 ú 11 ú 12ø 2 ú 21 ú 22 Σ[ÿ [0 τ[ø )\Σ

;ûØû[y = y9yL ]

z9u ] ;ûØû uL9 [u: w:x0:

x9

[x0:

[x9

Figure1.4: DPF-controller[Σ for Σ ï WPLSñ U 4 W ô H ô Z 4 Y ò

As in Section7.3, the output “y” is now divided in two, namely“y” í^ zy � ,wherez is theobjectiveoutputto beminimizedandy is ameasurementthatis fedinto thecontroller. Thiscorrespondsto thedynamics� ! " x@ í Ax ù B1u ù B2wôz í C1x ù D11u ù D12wô

y í C2x ù D21u ù D22wô (1.25)with initial state x0 ï H, disturbanceinput w ï L2 ñ R û ;W ò , control inputu ï L2 ñ R û ;U ò , objective output z ï L2 ñ R û ;Z ò and measurementoutput y ïL2 ñ R û ;Y ò (thecontrollerinput). In thecaseof a generalweaklyregularsystem,equations(1.25)hold in thestrongsense,seeTheorem6.2.13for details.

We arethento find a controller ) : y Aü u s.t. thenormw Aü z becomessmallenoughand that the closed-loopconnectionbecomesexponentially stable(thatis the maincase;we only treatthecasewherethe closed-loopsystemis merelyrequiredto bestableor strongly stable).

(We remindthattheorderof thesubindicescorrespondingto u andw is oftenreversedin theliterature;thisalsoaffectstheformulaebelow.)

In Section12.1, we presentseveral versionsof the standardresult that theH∞ 4BP has a solution if f the two H∞ Riccati equationshave nonnegativeexponentially stabilizingsolutionssatisfyingthecouplingcondition.Sincewedonot useany simplifying assumptions, our formulaebecomerathercomplicated.Therefore,we show here the simplified forms of thoseformulae (by makingadditionalassumptions):

Theorem 1.2.10(H∞H∞H∞ 4BP) Let γ � 0. Make the regularity and nonsingularityassumptions(A1)&(A2)of Theorem12.1.4.

Thenthere is anexponentiallystabilizing DPF-controller for Σ (possibly withinternal loop)satisfying � w Aü z �_� γ iff (1.)–(3.)of Theorem12.1.4hold. Underthenormalizingconditions

D12 í 0 í D21 ô D î11 èC1 D11ì í è 0 I ì í D22 / Bî2D î221 ô (1.26)conditions(1.)–(3.)canbewrittenasfollows:

(1.) (� X� X� X-CARE) There is � X ïUðñ H ô Domñ B îw ò�ò s.t. � X � 0 on H, A ù

38 CHAPTER1. INTRODUCTIONñ γ � 2B2 ñ Bî2 ò w E B1 ñ Bî1 ò w ò � X is exponentiallystable, andñ�ñ Bî1 ò w � X ò î ñ Bî1 ò w � X E γ � 2 ñ�ñ Bî2 ò w � X ò î ñ Bî2 ò w � X í Aî � X ù � XA ù C î1C1 �(1.27)

(2.) (� Y� Y� Y-CARE) There is � Y ï¶ðñ H ô Domñ F C2C1 > w ò�ò s.t. � Y � 0 on H, A î ùñ γ � 2C î1 ñ C1 ò w E C î2 ñ C î2 ò w ò � Y is exponentially stable, andñ�ñ C2 ò w � X ò î ñ C2 ò w � X E γ � 2 ñ�ñ C1 ò w � X ò î ñ C1 ò w � X í A� X ù � XAî ù B2Bî2 �(1.28)

(3.) (Coupling condition) ρ ñ � X � Y ò`� γ2.Any solutions of (1.) or (2.) are unique. If (1.)–(3.) are satisfied, then

all exponentiallystabilizing DPF-controllers for Σ satisfying � w Aü z �a� γ arethe onesparametrizedin Theorem12.1.8,and the regularity claimsof Theorem12.1.4(a)&(b)apply.

In (3.), ρ denotesthe spectral radius. One of the alternative regularityassumptions in (A1) is thatB is boundedandπ � 0 � 1 � Cw ÿµï L1 ñ 0 ô 1ò ; ð¯ñ H ô Z 4 Y ò�ò .For boundedB, theRiccatiequation(1.27)takestheclassicalform� X ñ B1Bî1 E γ � 2B2Bî2 ò � X í Aî � X ù � XA ù C î1C1 � (1.29)Seep. 618 for further simplification andremarks. Analogousremarksapply to(2.); e.g.,for boundedC, theRiccatiequation(1.28)becomes� X ñ C î2C2 E γ � 2C î1C1 ò � X í A� X ù � XAî ù B2Bî2 � (1.30)Thus, the classicalresultsbecomespecialcasesof ours. We also give sev-eral results under weaker regularity assumptions (e.g., for the case whereÿ B ô Cw ÿ ô Cw ÿ B ï L1loc; this allows roughlytwice asmuchunboundednessastheassumptions of aPritchard–Salamonsystem).

In generalwe allow for DPF-controllerswith internalloop, but we show thatsucha loop is not neededif D21 í 0 (i.e., onecanusea well-posedcontrollerinthatcase).

In Section12.2, we give discreteforms of the resultsof Chapter12. Forthemwe needno regularity assumptions(sinceB andC arealwaysboundedfor“discrete-timeWPLSs”).

In Section12.3,we studythe frequency-domainH∞ 4BP, whereoneis onlygivenan I/O map ú ï TIC∞ ñ U 4 W ô Z 4 Y ò , andonewishesto find a controller(I/O map) )Ìï TIC∞ ñ Yô U ò s.t. the closed-loopconnectionbecomes(I/O-)stableandsatisfies� w Aü z �_� γ (see(1.14)for bdcñ ú�ô )¹ò : w Aü z; we alsotreatthecasewhere ) is allowed to have an internal loop). In

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