Infinite-Dimensional Linear Systems, Optimal Control and Algebraic Riccati Equations

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Notation: oo = infinity, ^ = Laplace/Fourier transform, -> = "is mapped to".
Structure: The last number on most rows is the page number (to which you should add 0-5; all page numbers in the thesis outside the contents are correct). The printed version of the book is in three volumes. The book consists of parts I--IV, Appendices A--F and some supplements.

Abstract, keywords  (html)

Preface  (html)

C O N T E N T S

Volume 1/3

Abstract, contents, figures, preface, ...   (ps)  -1
Abstract, keywords  (html)   (pdf)   (ps)  2
Preface  (html)   (pdf)   (ps)  9

1  Introduction   (ps)  11
     1.1  On the contributions of this book 12
     1.2 A summary of this book  15
     1.3 Conventions  43

I    TI Operator Theory 45

2  TI and MTI Operators   (ps)  47
     2.1 Time-invariant operators (TI) 48
     2.2 GTIC  --  invertibility  55
     2.3 Static operators  63
     2.4 The signature operator S  65
     2.5 Losslessness  68
     2.6 MTI and its subclasses (measure convolutions)  71

3  Transfer Functions ($Lap\; TI\; =\; Loo$_strong, Lap TIC = H^oo)   (ps)  79
     3.1 Transfer functions of TI (Lap TI = L^oo_strong)  80
     3.2 Lap TI = L^oo_strong for Banach spaces (Fourier multipliers)  90
     3.3 H² and H^oo boundary functions in L² and L^oo_strong  99

4  Corona Theorems and Inverses   (ps)  115

5  Spectral Factorization (E=Y*X)   (ps)  133
     5.1 Auxiliary spectral factorization results  133
     5.2 MTI spectral factorization (E,X,Y in MTI)  140

II Continuous-Time Control Theory  149

6  Well-Posed Linear Systems (WPLS)   (ps)  151
     6.1 WPLS theory  153
     6.2 Regularity (^D(+oo) exists)  166
     6.3 Further regularity and compatibility  180
     6.4 Spectral and coprime factorizations (D=NM^-1)  202
     6.5 Further coprimeness and factorizations  210
     6.6 Feedback and Stabilization  219
     6.7 Further feedback results  244
     6.8 Systems with ABu₀ in L^p([0,1];H)  263
     6.9 Bounded B, bounded C, PS-systems  272

7  Dynamic Stabilization   (ps)  279
     7.1 Dynamic feedback (DF) stabilization   280
     7.2 DF-stabilization with internal loop  291
     7.3 DPF-stabilization (F_l(D , Q ))  314

Volume 2/3

III Riccati equations and Optimal control  347

8  Optimal Control (^d/_duJ=0)   (ps)  349
     8.1 Abstract J-critical control (Jy_ycrit _|_ Δy )  351
     8.2 Abstract J-coercivity (J -> [u Du])   356
     8.3 J-critical control for WPLSs   361
     8.4 J-coercivity and factorizations  377
     8.5 Problems on a finite time interval  392
     8.6 Extended linear systems (ELS)  395

9  Riccati Equations (ARE) and J-Critical Control   (ps)  401
     9.1 The Riccati Equation: A summary for U_out (r.c.f. <=> CARE)  404
     9.2 Riccati equations when ABu₀ in L¹  417
     9.3 Proofs for Section 9.2   432
     9.4 Analytic semigroups  438
     9.5 Parabolic problems and CAREs  443
     9.6 Parabolic problems: proofs  450
     9.7 Riccati equations on Dom(A_crit)  452
     9.8 Algebraic and integral Riccati equations (CARE <=> IARE)  465
     9.9 J-Critical control <=> Riccati Equation  481
     9.10 Proofs for Section 9.9: Crit <=> eIARE  500
     9.11 Proofs for Section 9.8: eCARE <=> eIARE  507
     9.12 Further eIARE and eCARE results  517
     9.13 Examples of Riccati equations  525
     9.14 (J,*)-critical factorization (D=NM^-1)  534
     9.15 H²-factorization when dim U < infty   539

10  Quadratic Minimization (min J)   (ps)  543
     10.1 Minimizing /₀^oo ||y||²+||u||²  (LQR)   545
     10.2 General minimization (LQR)  553
     10.3 Standard assumptions  568
     10.4 The H² problem  580
     10.5 Real lemmas  587
     10.6 Positive Popov operators (ε I < D*JD = X*X)  592
     10.7 Positive Riccati equations (J(0,u) >= 0)  601

11    H^oo Full-Information Control Problem   (||z||₂ =< γ||w||₂)   (ps)  607
     11.1 The H^oo Full-Info Control Problem (FICP)  608
     11.2 The H^oo FICP: proofs  625
     11.3 The H^oo FICP: stable case  649
     11.4 Minimax J-coercivity  665
     11.5 The discrete-time H^oo ficp   669
     11.6 The H^oo ficp: proofs  673
     11.7 The abstract H^oo FICP  676
     11.8 The Nehari problem  680
     11.9 The proofs for Section 11.8   682

12    H^oo Four-Block Problem (||F_l(D , Q )|| < γ)   (ps)  685
     12.1 The standard H^oo problem ( H^oo 4BP)  686
     12.2 The discrete-time H^oo problem ( H^oo 4bp)  705
     12.3 The frequency-space (I/O) H^oo 4BP  709
     12.4 Proofs for Section 12.3   719
     12.5 Proofs for Section 12.1  --  P_X, P_Y, P_Z  733
     12.6 Proofs for Section 12.2  --  4bp P_X, P_Y, P_Z  758

Volume 3/3

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

13  Discrete-Time Maps and Systems (ti & wpls)   (ps)  777
     13.1 Discrete-time I/O maps (tic)  779
     13.2 The Cayley transform  787
     13.3 Discrete-time systems (wpls(U,H,Y))  792
     13.4 Time discretization (Δ^S: WPLS -> wpls)  805

14  Riccati Equations (DARE)   (ps)  815
     14.1 Discrete-time Riccati equations (DARE)  815
     14.2 DARE --- further results  820
     14.3 Spectral and coprime factorizations  829

15  Quadratic Minimization   (ps)  833
     15.1 J-critical control and minimization  833
     15.2 Standard assumptions in discrete time  839
     15.3 Positive DAREs  842
     15.4 Real lemmas  843
     15.5 Riccati inequalities and the maximal solution  846

  Conclusions   (ps)   851

A  Algebraic and Functional Analytic Results   (ps)  853

A.1 Algebraic auxiliary results (Inverse(	A B 0 D	)   =   (	A-1 -A-1BD-1 0 D-1	))  854
A.2 Topological spaces  863     A.3 Hilbert and Banach spaces  865     A.4 C0-Semigroups  896

B  Integration and Differentiation in Banach Spaces   (ps)  903
     B.1 The Lebesgue integral and L^p(R;[0,+oo)) spaces  904
     B.2 Bochner measurability (L(Q;B))  906
     B.3 Lebesgue spaces (L^p(Q,μ;B))  912
     B.4 The Bochner integral (/_Q: L¹(Q;B) -> B)  922
     B.5 Differentiation of integrals  937
     B.6 Vector-valued distributions D '(Ω;B)  943
     B.7 Sobolev spaces W^k,p(Ω;B)  945

C  Almost Periodic Functions (AP)   (ps)   953

D  Laplace and Fourier Transforms (Lap u = ^u)   (ps)   957

E  Interpolation Theorems   (ps)   983
     E.1 Interpolation theorems (L^p1 + L^p2 --> L^q1 + L^q2)  983

F    L^p_strong, L^p_weak and Integration   (ps)   993
     F.1  L^p_strong, L^p_weak  993
     F.2 Strong and weak integration  1007
     F.3 Weak Laplace transform  1013

  References   (ps)  1020

  Notation   (ps)  1033
     Glossary 1044
     Abbreviations 1045
     Acronyms 1046

  Index   (ps)  1047