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Well-Posed Linear Systems

OLOF STAFFANS Department ofMathematics Abo˚ Akademi University, Finland

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Library ofCongress Cataloging in Publication data Staffans, Olof J., 1947– Well-posed linear systems / Olof Staffans. p. cm. – (Encyclopedia of mathematics and its applications) Includes bibliographical references and index. ISBN 0-521-82584-9 1. Linear systems. 2. System theory. I. Staffans, Olof J., 1947– II. Title. III. Series. QA372.S784 2004 003.74–dc22 2004051868

ISBN-13 978-0-521-82584-9 hardback ISBN-10 0-521-82584-9 hardback

Contents

List ofﬁgures page ix Preface xi List ofnotation xiv

1 Introduction and overview 1 1.1 Introduction 1 1.2 Overview of chapters 2–13 8 2 Basic properties of well-posed linear systems 28 2.1 Motivation 28 2.2 Deﬁnitions and basic properties 34 2.3 Basic examples of well-posed linear systems 46 2.4 Time discretization 55 2.5 The growth bound 60 2.6 Shift realizations 67 2.7 The Lax–Phillips scattering model 71 2.8 The Weiss notation 76 2.9 Comments 78 3 Strongly continuous semigroups 85 3.1 Norm continuous semigroups 85 3.2 The generator of a C0 semigroup 87 3.3 The spectra of some generators 98 3.4 Which operators are generators? 106 3.5 The dual semigroup 113 3.6 The rigged spaces induced by the generator 122 3.7 Approximations of the semigroup 128 3.8 The nonhomogeneous Cauchy problem 133 3.9 Symbolic calculus and fractional powers 140 3.10 Analytic semigroups and sectorial operators 150

vi Contents

3.11 Spectrum determined growth 164 3.12 The Laplace transform and the frequency domain 169 3.13 Shift semigroups in the frequency domain 177 3.14 Invariant subspaces and spectral projections 180 3.15 Comments 191 4 The generators of a well-posed linear system 194 4.1 Introduction 194 4.2 The control operator 196 4.3 Differential representations of the state 202 4.4 The observation operator 213 4.5 The feedthrough operator 219 4.6 The transfer function and the system node 227 4.7 Operator nodes 238 4.8 Examples of generators 256 4.9 Diagonal and normal systems 260 4.10 Decompositions of systems 266 4.11 Comments 273 5 Compatible and regular systems 276 5.1 Compatible systems 276 5.2 Boundary control systems 284 5.3 Approximations of the identity in the state space 295 5.4 Extended observation operators 302 5.5 Extended observation/feedthrough operators 313 5.6 Regular systems 317 5.7 Examples of regular systems 325 5.8 Comments 329 6 Anti-causal, dual, and inverted systems 332 6.1 Anti-causal systems 332 6.2 The dual system 337 6.3 Flow-inversion 349 6.4 Time-inversion 368 6.5 Time-ﬂow-inversion 378 6.6 Partial ﬂow-inversion 386 6.7 Comments 400 7 Feedback 403 7.1 Static output feedback 403 7.2 Additional feedback connections 413 7.3 State feedback and output injection 422 7.4 The closed-loop generators 425

Contents vii

7.5 Regularity of the closed-loop system 433 7.6 The dual of the closed-loop system 436 7.7 Examples 436 7.8 Comments 440 8 Stabilization and detection 443 8.1 Stability 443 8.2 Stabilizability and detectability 453 8.3 Coprime fractions and factorizations 465 8.4 Coprime stabilization and detection 473 8.5 Dynamic stabilization 485 8.6 Comments 502 9Realizations 505 9.1 Minimal realizations 505 9.2 Pseudo-similarity of minimal realizations 511 9.3 Realizations based on factorizations of the Hankel operator 517 9.4 Exact controllability and observability 521 9.5 Normalized and balanced realizations 530 9.6 Resolvent tests for controllability and observability 538 9.7 Modal controllability and observability 546 9.8 Spectral minimality 549 9.9 Controllability and observability of transformed systems 551 9.10 Time domain tests and duality 554 9.11 Comments 565 10 Admissibility 569 10.1 Introduction to admissibility 569 10.2 Admissibility and duality 572 10.3 The Paley–Wiener theorem and H ∞ 576 10.4 Controllability and observability gramians 583 10.5 Carleson measures 591 10.6 Admissible control and observation operators for diagonal and normal semigroups 598 10.7 Admissible control and observation operators for contraction semigroups 602 10.8 Admissibility results based on the Lax–Phillips model 610 10.9 Comments 613 11 Passive and conservative scattering systems 616 11.1 Passive systems 616 11.2 Energy preserving and conservative systems 628 11.3 Semi-lossless and lossless systems 636

viii Contents

11.4 Isometric and unitary dilations of contraction semigroups 643 11.5 Energy preserving and conservative extensions of passive systems 655 11.6 The universal model of a contraction semigroup 660 11.7 Conservative realizations 670 11.8 Energy preserving and passive realizations 677 11.9 The Spectrum of a conservative system 691 11.10 Comments 692 12 Discrete time systems 696 12.1 Discrete time systems 696 12.2 The internal linear fractional transform 703 12.3 The Cayley and Laguerre transforms 707 12.4 The reciprocal transform 719 12.5 Comments 728 Appendix 730 A.1 Regulated functions 730 A.2 The positive square root and the polar decomposition 733 A.3 Convolutions 736 A.4 Inversion of block matrices 744

Bibliography 750 Index 767

Figures

2.1 Regular well-posed linear system page 29 2.2 Well-posed linear system 38 2.3 Cross-product (the union of two independent systems) 52 2.4 Cross-product in block form 52 2.5 Sum junction 53 2.6 Sum junction in block form 53 2.7 T-junction 54 2.8 T-junction in block form 54 2.9 Parallel connection 55 2.10 Parallel connection in block form 55 3.1 The sector δ 150 3.2 The sector θ,γ 151 3.3 The path in the proof of Theorem 3.10.5 153 7.1 Static output feedback connection 404 7.2 Positive identity feedback 409 7.3 Negative identity feedback 410 7.4 Flow-inversion 412 7.5 Another static output feedback 413 7.6 A third static output feedback 413 7.7 Output feedback in block form 415 7.8 Cancellation of static output feedback 417 7.9 Cascade connection through K 417 7.10 Cascade connection in block form 418 7.11 Dynamic feedback 419 7.12 Dynamic feedback in block form 419 7.13 Partial feedback 421 7.14 State feedback 422 7.15 Output injection 423 7.16 Original system with one extra output 426

x Figures

7.17 Closed-loop system with one extra output 426 8.1 The extended system 453 8.2 Right coprime factor 454 8.3 Left coprime factor 455 8.4 Cancellation of state feedback 456 8.5 Dynamic stabilization 485 8.6 Dynamic stabilization 494 8.7 Equivalent version of dynamic stabilization 495 8.8 Second equivalent version of dynamic stabilization 497 8.9 Third equivalent version of dynamic stabilization 498 8.10 Youla parametrization 499 8.11 Youla parametrized stabilizing compensator 500 8.12 Youla parametrized stabilizing compensator 501 8.13 Youla parametrized stabilizing compensator 501

Preface

This main purpose of this book is to present the basic theory of well-posed linear systems in a form which makes it available to a larger audience, thereby opening up the possibility of applying it to a wider range of problems. Up to now the theory has existed in a distributed form, scattered between different papers with different (and often noncompatible) notation. For many years this has forced authors in the field (myself included) to start each paper with a long background section to first bring the reader up to date with the existing theory. Hopefully, the existence of this monograph will make it possible to dispense with this in future. My personal history in the field of abstract systems theory is rather short but intensive. It started in about 1995 when I wanted to understand the true nature of the solution of the quadratic cost minimization problem for a linear Volterra integral equation. It soon became apparent that the most appropriate setting was not the one familiar to me which has classically been used in the field of Volterra integral equations (as presented in, e.g., Gripenberg et al. [1990]). It also became clear that the solution was not tied to the class of Volterra integral equations, but that it could be formulated in a much more general framework. From this simple observation I gradually plunged deeper and deeper into the theory of well-posed (and even non-well-posed) linear systems. One of the first major decisions that I had to make when I began to write this monograph was how much of the existing theory to include. Because of the nonhomogeneous background of the existing theory (several strains have been developing in parallel independently of each other), it is clear that it is impossible to write a monograph which will be fully accepted by every worker in the field. I have therefore largely allowed my personal taste to influence the final result, meaning that results which lie closer to my own research interests are included to a greater extent than others. It is also true that results which blend more easily into the general theory have had a greater chance of being included than those which are of a more specialist nature. Generally speaking,

xii Preface

instead of borrowing results directly from various sources I have reinterpreted and reformulated many existing results into a coherent setting and, above all, using a coherent notation. The original motivation for writing this book was to develop the background which is needed for an appropriate understanding of the quadratic cost minimization problem (and its indefinite minimax version). However, due to page and time limitations, I have not yet been able to include any optimal control in this volume (only the background needed to attack optimal control problems). The book on optimal control still remains to be written. Not only was it difficult to decide exactly what parts of the existing theory to include, but also in which form it should be included. One such decision was whether to work in a Hilbert space or in a Banach space setting. Optimal control is typically done in Hilbert spaces. On the other hand, in the basic theory it does not matter if we are working in a Hilbert space or a Banach space (the technical differences are minimal, compared to the general level of difficulty of the theory). Moreover, there are several interesting applications which require the use of Banach spaces. For example, the natural norm in population dynamics is often the L1-norm (representing the total mass), parabolic equations have a well-developed L p-theory with p = 2, and in nonlinear equations it is often more convenient to use L∞-norms than L2-norms. The natural decision was to present the basic theory in an arbitrary Banach space, but to specialize to Hilbert spaces whenever this additional structure was important. As a consequence of this decision, the present monograph contains the first comprehensive treatment of a well-posed linear system in a setting where the input and output signals are continuous (as opposed to belonging to some L p-space) but do not have any further differentiability properties (such as belonging to some Sobolev spaces). (More precisely, they are continuous apart from possible jump discontinuities.) The first version of the manuscript was devoted exclusively to well-posed problems, and the main part of the book still deals with problems that are well posed. However, especially in H ∞-optimal control, one naturally runs into non- well-posed problems, and this is also true in circuit theory in the impedance and transmission settings. The final incident that convinced me that I also had to include some classes of non-well-posed systems in this monograph was my discovery in 2002 that every passive impedance system which satisfies a certain algebraic condition can be represented by a (possibly non-well-posed) system node. System nodes are a central part of the theory of well-posed systems, and the well-posedness property is not always essential. My decision not to stay strictly within the class of well-posed systems had the consequence that this monograph is also the the first comprehensive treatment of (possibly non-well- posed) systems generated by arbitrary system nodes.

Preface xiii

The last three chapters of this book have a slightly different ﬂavor from the earlier chapters. There the general Banach space setting is replaced by a stan- dard Hilbert space setting, and connections are explored between well-posed linear systems, Fourier analysis, and operator theory. In particular, the admissibility of scalar control and observation operators for contraction semigroups is characterized by means of the Carleson measure theorem, and systems theory interpretations are given of the basic dilation and model theory for contractions and continuous-time contraction semigroups in Hilbert spaces. It took me approximately six years to write this monograph. The work has primarily been carried out at the Mathematics Institute of Abo˚ Akademi, which has offered me excellent working conditions and facilities. The Academy of Finland has supported me by relieving me of teaching duties for a total of two years, and without this support I would not have been able to complete the manuscript in this amount of time. I am grateful to several students and colleagues for helping me ﬁnd errors and misprints in the manuscript, most particularly Mikael Kurula, Jarmo Malinen and Kalle Mikkola. Above all I am grateful to my wife Marjatta for her understanding and patience while I wrote this book.

Notation

Basic sets and symbols

C The complex plane. + + + + Cω , Cω Cω :={z ∈ C | z >ω} and Cω :={z ∈ C | z ≥ ω}. − − − − Cω , Cω Cω :={z ∈ C | z <ω} and Cω :={z ∈ C | z ≤ ω}. C+, C+ C+ = C+ C+ = C+ : 0 and : 0 . C−, C− C− = C− C− = C− : 0 and : 0 . D+, D+ D+ ={ ∈ C | > } D+ ={ ∈ C || |≥ } r r r : z z r and r : z z r . D−, D− D− ={ ∈ C | < } D− ={ ∈ C || |≤ } r r r : z z r and r : z z r . D+, D+ D+ = D+ D+ = D+ : 1 and : 1 . D−, D− D− = D− D− = D− : 1 and : 1 . RR:= (−∞, ∞). + + R+, R R+ := (0, ∞) and R := [0, ∞). − − R−, R R− := (−∞, 0) and R := (−∞, 0]. T The unit circle in the complex plane. TT The real line R where the points t + mT, m = 0, ±1, ±2,... are identiﬁed. Z The set of all integers. Z+ − + − , Z Z :={√ 0, 1, 2,...} and Z :={−1, −2, −3,...}. jj:= −1. 0 The number zero, or the zero vector in a vector space, or the zero operator, or the zero-dimensional vector space {0}. 1 The number one and also the identity operator on any set.

Operators and related symbols

A, B, C, D In connection with an L p|Reg-well-posed linear system or an operator node, A is usually the main operator, B the control

xiv

Notation xv

operator, C the observation operator and D a feedthrough operator. See Chapters 3 and 4. C&D The observation/feedthrough operator of an L p|Reg-well- posed linear system or an operator node. See Definition 4.7.2. A, B, C, D The semigroup, input map, output map, and input/output map of an L p|Reg-well-posed linear system, respectively. See Def- initions 2.2.1 and 2.2.3. D The transfer function of an L p|Reg-well-posed linear system or an operator node. See Definitions 4.6.1 and 4.7.4. B(U; Y ), B(U) The set of bounded linear operators from U into Y or from U into itself, respectively. C, L The Cayley and Laguerre transforms. See Definition 12.3.2. τ t The bilateral time shift operator τ t u(s):= u(t + s) (this is a left-shift when t > 0 and a right-shift when t < 0). See Example 2.5.3 for some additional shift operators. γλ The time compression or dilation operator (γλu)(s):= u(λs). Here λ>0. πJ (πJ u)(s):= u(s)ifs ∈ J and (πJ u)(s):= 0ifs ∈/ J. Here J ⊂ R. π+,π− π+ := π[0,∞) and π− := π(−∞,0). R The time reflection operator about zero: ( R u)(s):= u(−s) p (in the L -case) or ( R u)(s):= limt↓−s u(t) (in the Reg-case). See Definition 3.5.12. R h The time reflection operator about the point h. See Lemma 6.1.8. σ The discrete-time bilateral left-shift operator (σu)k := uk+1, where u ={uk }k∈Z. See Section 12.1 for the definitions of σ+ and σ−. πJ (πJ u)k := uk if k ∈ J and (πJ u)k := 0ifk ∈/ J. Here J ⊂ Z and u ={uk }k∈Z. π+, π− π+ := πZ+ and π− := πZ− . w-lim The weak limit in a Banach space. Thus w-limn→∞ xn = x in ∗ ∗ ∗ ∗ X iff limn→∞ x xn = x x for all x ∈ X . See Section 3.5. x, x∗ In a Banach space setting x∗x :=x, x∗ is the continuous linear functional x∗ evaluated at x. In a Hilbert space setting this is the inner product of x and x∗. See Section 3.5. E⊥ E⊥ :={x∗ ∈ X ∗ |x, x∗=0 for all x ∈ E}. This is the annihilator of E ⊂ X. See Lemma 9.6.4. ⊥ F ⊥ F :={x ∈ X |x, x∗=0 for all x∗ ∈ F}. This is the pre- annihilator of F ⊂ X ∗. See Lemma 9.6.4. In the reflexive case ⊥ F = F ⊥, and in the nonreflexive case ⊥ F = F ⊥ ∩ X.

xvi Notation

A∗ The (anti-linear) dual of the operator A. See Section 3.5. A ≥ 0 A is (self-adjoint and) positive definite. A 0 A ≥ for some >0, hence A is invertible. D (A) The domain of the (unbounded) operator A. R (A) The range of the operator A. N (A) The null space (kernel) of the operator A. rank(A) The rank of the operator A. dim(X) The dimension of the space X. ρ(A) The resolvent set of A (see Definition 3.2.7). The resolvent set is always open. σ(A) The spectrum of A (see Definition 3.2.7). The spectrum is always closed. σp(A) The point spectrum of A, or equivalently, the set of eigenval- ues of A (see Definition 3.2.7). σr (A) The residual spectrum of A (see Definition 3.2.7). σc(A) The continuous spectrum of A (see Definition 3.2.7). ωA The growth bound of the semigroup A. See Definition 2.5.6. TI, TIC T I stands for the set of all time-invariant, and TIC stands for the set of all time-invariant and causal operators. See Defini- tion 2.6.2 for details. A&B, C&DA&B stands for an operator (typically unbounded) whose D X domain (A&B) is a subspace of the cross-product U of two Banach spaces X and U, and whose values lie in a third Banach space Z.IfD (A&B) splits into D (A&B) = X1 +˙ U1 where X1 ⊂ X and U1 ⊂ U, then A&B can be written in = = block matrix form as A&B [AB], where A A&B|X1 = | and B A&B U1 . We alternatively write these identities in = x = 0 the form Ax A&B 0 and Bu A&B u , interpreting D (A&B) as the cross-product of X1 and U1.

Special Banach spaces

U Frequently the input space of the system. X Frequently the state space of the system. Y Frequently the output space of the system. Xn Spaces constructed from the state space X with the help of the generator of a semigroup A. In particular, X1 is the domain of the semigroup generator. See Section 3.6. ∗ ∗ = ∗ = ∗ Xn Xn : (X )n (X−n) . See Remark 3.6.1. +˙ X = X1 +˙ X2 means that the Banach space X is the direct sum of X1 and X2, i.e., both X1 and X2 are closed subspaces

Notation xvii

of X, and every x ∈ X has a unique representation of the form x = x1 + x2 where x1 ∈ X1 and x2 ∈ X2. ⊕ X = X1 ⊕ X2 means that the Hilbert space X is the or- thogonal direct sum of the Hilbert spaces X1 and X2, i.e., = + ⊥ X X1 ˙ X2 and X1 X2. X Y The cross-product of the two Banach spaces X and Y . Thus, X = X +˙ 0 Y 0 Y .

Special functions

χI The characteristic function of the set I . 1+ The Heaviside function: 1+ = χR+ . Thus (1+)(t) = 1 for t ≥ 0 and (1+)(t) = 0 for t < 0. B The Beta function (see (5.3.1)). The Gamma function (see (3.9.7)). ωt eω eω(t) = e for ω, t ∈ R. log The natural logarithm.

Function spaces

V (J; U) Functions of type V (= L p, BC, etc.) on the interval J ⊂ R with range in U. Vloc(J; U) Functions which are locally of type V , i.e., they are deﬁned on J ⊂ R with range in U and they belong to V (K ; U) for every bounded subinterval K ⊂ J. Vc(J; U) Functions in V (J; U) with bounded support. Vc,loc(J; U) Functions in Vloc(J; U) whose support is bounded to the left. Vloc,c(J; U) Functions in Vloc(J; U) whose support is bounded to the right. V0(J; U) Functions in V (J; U) vanishing at ±∞. See also the special cases listed below. −ωt Vω(J; U) The set of functions u for which (t → e u(t)) ∈ V (J; U). See also the special cases listed below. Vω,loc(R; U) The set of functions u ∈ Vloc(R; U) which satisfy π−u ∈ − Vω(R ; U). V (TT ; U) The set of T -periodic functions of type V on R. The norm in this space is the V -norm over one arbitrary interval of length T . BC Bounded continuous functions; sup-norm. BC0 Functions in BC that tend to zero at ±∞. −ωt BCω Functions u for which (t → e u(t)) ∈ BC.

xviii Notation

− BCω,loc(R; U) Functions u ∈ C(R; U) which satisfy π−u ∈ BCω(R ; U). −ωt BC0,ω Functions u for which (t → e u(t)) ∈ BC0. − BC0,ω,loc(R; U) Functions u ∈ C(R; U) which satisfy π−u ∈ BC0,ω(R ; U). BUC Bounded uniformly continuous functions; sup-norm. BUCn Functions which together with their n first derivatives belong to BUC. See Definition 3.2.2. C Continuous functions. The same space as BCloc. Cn n times continuously differentiable functions. The same n space as BCloc. C∞ Infinitely many times differentiable functions. The same ∞ space as BCloc. /p L p, 1 ≤ p < ∞ Strongly measurable functions with norm |u(t)|p dt 1 . L∞ Strongly measurable functions with norm ess sup|u(t)|. p p = p ≤ < ∞ ∞ ∈ ∞ L0 L0 L if 1 p , and L0 consists of those u L ±∞ | |= which vanish at , i.e., limt→∞ ess sup|s|≥t u(s) 0. p −ω Lω Functions u for which (t → e t u(t)) ∈ L p. p R ∈ p R π ∈ p R− Lω,loc( ; U) Functions u Lloc( ; U) which satisfy −u Lω( ; U). p → −ωt ∈ p L0,ω Functions u for which (t e u(t)) L0 . p R ∈ p R π ∈ p R− L0,ω,loc( ; U) Functions u Lloc( ; U) which satisfy −u L0,ω( ; U). W n,p Functions which together with their n first (distribution) derivatives belong to L p. See Definition 3.2.2. Reg Bounded right-continuous functions which have a left hand limit at each finite point. ±∞ Reg0 Functions in Reg which tend to zero at . −ωt Regω The set of functions u for which (t → e u(t)) ∈ Reg. R ∈ R π ∈ Regω,loc( ; U) The set of functions u Regloc( ; U) which satisfy −u − Regω(R ; U). → −ωt ∈ Reg0,ω The set of functions u for which (t e u(t)) Reg0. R ∈ R π ∈ Reg0,ω,loc( ; U) Functions u Regloc( ; U) which satisfy −u R− Reg0,ω( ; U). Regn Functions which together with their n first derivatives belong to Reg. See Definition 3.2.2. L p|Reg This stands for either L p or Reg, whichever is appropriate.