Systems & Control: Foundations & Applications

Series Editor

Tamer Ba~ar, University of Illinois at Urbana-Champaign

Editorial Board

Karl Johan Astrom, Lund Institute of Technology, Lund Han-Fu Chen, Academia Sinica, Bejing William Helton, University of California, San Diego Alberto Isidori, University of Rome and

Washington University, St. Louis Petar V. Kokotovic, University of California, Santa Barbara Alexander Kurzhanski, Russian Academy of Sciences, Moscow and

University of California, Berkeley H. Vincent Poor, Princeton University Mete Soner, K09 University, Istanbul

Hisham Abou-Kandil Gerhard Freiling Vlad Ionescu (t) Gerhard Jank

Matrix Riccati Equations in Control and Systems Theory

Springer Basel AG

Authors:

Hisham Abou-Kandil Ecole Normale Superieure de Cachan Laboratoire S.A.T.I.E. (UMR CNRS 8029) 61, Avenue du President Wilson F-94230 Cachan France hisham@satie.ens-cachan.fr

Gerhard Jank Institut fiir Mathematik RWTHAachen Templergraben 55 D-52056 Aachen jank@math2.rwth-aachen.de

Gerhard Freiling Institute of Mathematics University of Duisburg Lotharstrasse 65 D-47048 Duisburg Germany freiling@math.uni-duisburg.de

Vlad Ionescu (t) Department of Mathematics University of Bucharest Romania

2000 Mathematics Subject Classification 49N05, 49NlO, 49N35, 49N40, 49Nff, 90D05, 90D06, 90DI0, 90D50, 90D65, 90D25, 93B36, 93B52, 93B35,93E20, 15A24, 15A57, 34Cll, 34K35, 47B35

A CIP catalogue record for this book is available from the Library of Congress, Washington D.e., USA

Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>.

ISBN 978-3-0348-9432-6 ISBN 978-3-0348-8081-7 (eBook) DOI 10.1007/978-3-0348-8081-7

This work is subject to copyright. All rights are reserved, whether the whole or part ofthe material is concerned, specifically the rights oftranslation, reprinting, re-use of illustrations, re citation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permis sion of the copyright owner must be obtained.

© 2003 Springer Basel AG Originally published by Birkhăuser Verlag in 2003 Softcover reprint ofthe hardcover lst edition 2003

Printed on acid-free paper produced of chlorine-free pulp. rCF 00

ISBN 978-3-0348-9432-6

987654321

To the late V. Ionescu, our esteemed colleague, and

to our beloved families.

Contents

Preface xi

Notation xvi

1 Basic results for linear equations 1 1.1 Linear differential equations and linear algebraic equations. 1 1.2 Exponential dichotomy and L2 evolutions . . . . . . . . . . 11

2 Hamiltonian Matrices and Algebraic Riccati equations 21 2.1 Solutions of algebraic Riccati equations and graph subspaces 22 2.2 Indefinite scalar products and a canonical form of Hamiltonian

matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3 Hermitian algebraic Riccati equations ............. 50 2.4 Positive semi-definite solutions of standard algebraic Riccati

equations . . . . . . . . . . . . . . . . . . . . . . . . 58 2.5 Hermitian discrete-time algebraic Riccati equations. . . 80

3 Global aspects of Riccati differential and difference equations 89 3.1 Riccati differential equations and associated linear systems. 90

3.1.1 Riccati differential equations, Riccati-transformation and spectral factorization . . . . . . . . . . . . . . . 92

3.1.2 Riccati differential equations and linear boundary value problems ..................... 95

3.2 A representation formula. . . . . . . . . . . . . . . . . 97 3.3 Flows on GraBmann manifolds: The extended Riccati

differential equation .......................... 110 3.4 General representation formulae for solutions of RDE and PRDE,

the time-continuous and periodic Riccati differential equation, and dichotomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 3.4.1 A general representation formula for solutions of RDE . . . 134 3.4.2 A representation formula for solutions of the periodic Riccati

differential equation PRDE . . . . . . . . . . . . . . . . . . 144

viii Contents

3.5 A representation formula for solutions of the discrete time Riccati equation . . . . . . . . . . . . . . . . . . . . 150 3.5.1 Properties of the solutions to DARE 156 3.5.2 Properties of the solutions to DRDE 158

3.6 Global existence results ..... . 163

4 Hermitian Riccati differential equations 181 181 181 188 189 193

4.1 Comparison results for HRDE . 4.1.1 Arbitrary coefficients . 4.1. 2 Periodic coefficients 4.1.3 Constant coefficients . 4.1.4 Riccati inequalities . .

4.2 Monotonicity and convexity results: A Frechet derivative based approach 4.2.1 Notation and preliminaries 4.2.2 Results for HARE . 4.2.3 Results for HDARE ... . 4.2.4 Results for HRDE .... .

4.3 Convergence to the semi-stabilizing solution 4.4 Dependence of HRDE on a parameter . 4.5 An existence theorem for general HRDE 4.6 A special property of HRDE ...... .

198 198 202 206 207 209 220 239 247

5 The periodic Riccati equation 257 5.1 Linear periodic differential equations . . . . . . . . . . . . . 257 5.2 Preliminary notation and results for linear periodic systems 262 5.3 Existence results for periodic Hermitian Riccati equations 5.4 Positive semi-definite periodic equilibria of PRDE .

268 279

6 Coupled and generalized Riccati equations 299 6.1 Some basic concepts in dynamic games. . . . . . . . . . . . . . 299 6.2 Non-symmetric Riccati equations in open loop Nash differential

6.3 6.4

6.5 6.6 6.7

games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 Discrete-time open loop Nash Riccati equations . . . . . . . 313 Non-symmetric Riccati equations in open loop Stackelberg differential games . . . . . . . . . . . . . . . . . . . . . . . . 322 Discrete-time open loop Stackelberg equations. . . . . . . . 328 Coupled Riccati equations in closed loop Nash differential games 333 Rational matrix differential equations arising in stochastic control. 343

6.8 Rational matrix difference equations arising in stochastic control 377 6.9 Coupled Riccati equations in Markovian jump systems . . . . . . . 394

Contents IX

7 Symmetric differential Riccati equations: an operator based approach 411 7.1 Popov triplets: definition and equivalence 412 7.2 Associated objects . . . . . . . . . . 413 7.3 Associated operators . . . . . . . . . 422 7.4 Existence of the stabilizing solution. 428 7.5 Positivity theory and applications. 434 7.6 Differential Riccati inequalities . . . 439 7.7 The signature condition . . . . . . . 444 7.8 Differential Riccati theory: A Hamiltonian descriptor operator

approach. . . . . . . . . . . . . . . 455 7.8.1 Descriptors and dichotomy ........ 455 7.8.2 Hamiltonian descriptors . . . . . . . . . . 461 7.8.3 The stabilizing (anti-stabilizing) solution. 463

8 Applications to Robust Control Systems 467 8.1 The Four Block Nehari Problem . 467

8.1.1 Problem Statement . . . . . 467

8.2

8.1.2 A characterization of all solutions. 8.1.3 Main Result. . . . Disturbance Attenuation . . . 8.2.1 Problem statement .. 8.2.2 A necessary condition 8.2.3 The Disturbance Feedforward Problem. 8.2.4 The least achievable tolerance of the DF problem .

9 Non-symmetric Riccati theory and applications 9.1 Non-symmetric Riccati theory ...... .

9.1.1 Basic notions and preliminary results ... 9.1.2 Toeplitz operators and Riccati equations.

9.2 Application to open loop Nash games 9.2.1 Definitions and Hilbert space 9.2.2 Unique Nash equilibria .... . 9.2.3 The general case ....... . 9.2.4 If any, then one or infinitely many

9.3 Application to open loop Stackelberg games 9.3.1 Characterization in Hilbert space 9.3.2 Unique Stackelberg equilibria . 9.3.3 A value function type approach .

A Appendix A.1 Basic facts from control theory A.2 The implicit function theorem.

References

469 471 479 479 481 485 490

495 496 496 501 507 507 509 512 513 515 516 517 520

527 527 530

533

x

Index

List of Figures

Contents

569

572

Preface

In many fields of applied mathematics, engineering and economic sciences there appear, e.g., as a consequence of variational problems to be solved, matrix (or operator) Riccati equations. These can be algebraic Riccati equations as well as Riccati difference or differential equations. Riccati equations are in our opinion the simplest but most important class of non-linear equations. They show up in the following domains, just to cite a few:

- linear optimal control and filtering problems with quadratic cost functionals, - linear dynamic games with quadratic cost functionals, - decoupling of linear systems of differential and difference equations, - spectral factorization of operators, - singular perturbation theory, - boundary value problems for systems of ODEs, - invariant embedding and scattering theory, - differential geometry.

Besides the well-developed theory of symmetric (or Hermitian) Riccati equa­tions there are more and more demands from applications to develop further the theory for non-symmetric matrix Riccati equations and also for generalized or perturbed Riccati equation.

In the last decades developments of both algebraic and differential matrix Riccati equations theory spread out in the scientific literature. In contrast to the theory of symmetric (or hermitian) matrix algebraic Riccati equations, which has been presented recently in the book of Lancaster and Rodman [LaRo95], the ba­sic theory of matrix Riccati differential equations is comprised in the monograph of Reid [Reid72], along with some applications that were developed before 1972. During the last three decades there was achieved great progress in the math­ematical theory of Riccati equations and in its applications, with emphasis on control systems and differential games. Whereas symmetric Riccati equations play a central role in optimal control, non-symmetric matrix Riccati equations show up for instance in the theory of dynamic games and spectral factorization problems, while generalized Riccati equations are common in stochastic control problems or stochastic games.

The aim of this book is to present the state of the art of the theory of sym­metric (Hermitian) matrix Riccati equations and to contribute to the development of the theory of non-symmetric Riccati equations as well as to certain classes of coupled and generalized Riccati equations occurring in differential games and stochastic control. For the major part of the results presented in this book there exist infinite dimensional counterparts for operator Riccati equations; we do not address this topic here and refer the reader to the textbooks [BdPDM92] and [LaTr91]' [LaTrOOa], [LaTrOOb].

xii Preface

The book is intended to make available classical and recent results to engi­neers and mathematicians as well. Therefore we resign to present the results in their most general form, in particular we confine ourselves to the finite dimensional (i.e., matrix case). The book is accessible to graduate students in mathematics, applied mathematics, control engineering, physics or economics. It is written in an adequate mathematical style, high-lighting the main results so that it is easily accessible to researchers working in any of the fields where Riccati equations are used. It can also be recommended to engineers interested in the background for creating algorithms to compute control laws based on solving generalized Riccati equations.

The book is divided into nine chapters. Section 1.1 serves as an introduction to Chapters 2-6, containing basic facts on linear differential equations, in partic­ular on Lyapunov and Sylvester differential equations and on algebraic Lyapunov and Stein equations. Section 1.2 contains a short review of several basic facts in linear time-varying systems theory that are not readily available in the literature, such as dichotomy, £2 input-output maps, associated Toeplitz and Hankel oper­ators, nodes and duality. These concepts are the cornerstone for the theory of Riccati equations with time-varying coefficients developed in Chapters 7 and 8 and also for the non-symmetric generalization in Chapter 9.

In Chapter 2 we show that there is a one-to-one correspondence between the n-dimensional graph subspaces of an (m+n) x (m+n)-matrix M and the solutions of the associated non-symmetric algebraic Riccati equation. Moreover, if M = H is a Hamiltonian matrix, it is shown that the Hermitian solutions of the correspond­ing Hermitian algebraic Riccati equation HARE correspond to the Lagrangian invariant graph subspaces of H. Therefore we describe in Section 2.2 a canonical form of Hamiltonian matrices and provide a detailed description of its J-neutral and Lagrangian subspaces. These facts are used subsequently to describe the es­sential results on Hermitian solutions of HARE and of all positive semi-definite solutions of CARE, the special case of HARE appearing in linear quadratic control problems. In Section 2.5 we sketch how the Hermitian solutions of discrete-time Hermitian algebraic Riccati equations DARE can be derived from the Hermitian solutions of a corresponding continuous-time algebraic Riccati equation; this fact shows that all results derived for HARE can be translated into the corresponding results for DARE.

Chapter 3 then deals with global aspects and dynamic behavior of solutions of Riccati differential and difference equations. A general representation formula for all solutions is used to study the phase portrait and to obtain results on the possible number of solutions of the algebraic Riccati equation. The importance of exponential and also of polynomial dichotomy is explained. Global existence results for non-symmetric Riccati equations are presented, ensuring that a solution of an initial (terminal) value problem does not show finite escape time phenomenon, i.e., it does not blow up in finite time.

Preface xiii

In Chapter 4, we describe comparison results for different Riccati equations which are of Hermitian type. Also monotonicity and convexity properties are proved, using either the traditional approach or Frechet derivatives and the im­plicit function theorem. Moreover, we derive in Chapter 4 several basic existence, convergence and order preserving properties of the solutions of Hermitian Riccati differential equations, which are based on general comparison theorems.

The theory of periodic Hermitian Riccati differential equations is developed in Chapter 5. The approach used and the results obtained are parallel to the time invariant case. For convenience of the reader we give full proofs, moreover, in the first two sections we present some basic facts about periodic linear differential equations and we summarize the basic definitions and results from periodic linear control theory.

Chapter 6 is devoted to coupled and non-symmetric Riccati equations as­sociated to some problems in dynamic games and stochastic control. First open loop Nash and Stackelberg strategies are considered and the general represen­tation formula obtained in Chapter 3 is used to give a common framework to deal with non-symmetric Riccati equations, to prove existence and to study the asymptotic behavior of the solutions of the differential equations. Also, closed loop Nash-Riccati type equations are examined and global existence results are stated. Sections 6.7-6.9 contain recent results on generalized or perturbed Riccati equa­tions associated with some stochastic control problems, in particular it is shown that, for the solutions of the rational matrix equations appearing in stochastic con­trol, several of the nice properties of standard Riccati equations are maintained. Numerical algorithms for solving such type of equations are described.

Chapter 7 gives the main existence results for the stabilizing solution and other solutions related to the symmetric Riccati differential equation on an in­finite time horizon, as it appears in many branches of control theory. The main novelty is that we remove the restrictive assumptions usually present in the litera­ture and deal with the general case of an indefinite sign quadratic term, as appears in non-cooperative games, optimal control, approximation theory, and robust con­trol. In the first three sections we introduce the basic setting for our approach to the Riccati equation theory. The central notion here is the Popov triplet, for which we define a special equivalence transformation and with which we associate a number of mathematical objects that will play various parts: the Hamiltonian system, the Popov operator, the Hermitian Riccati differential system, the Her­mitian Riccati differential equation, and the Kalman-Popov-Yakubovich system. The key result of Section 7.4 says that the Hermitian Riccati differential equation has a stabilizing solution if and only if a family of Toeplitz operators associated with the underlying Hamiltonian system is uniformly boundedly invertible. In Sections 7.5 and 7.7 we refine further this result to give easily checkable solvabil­ity conditions in two important cases: the sign definite and the sign indefinite.

XIV Preface

We present also some relevant applications of the sign definite case in control systems, including the Bounded Real Lemma, The Kalman-Popov-Yakubovich Lemma and the Small Gain Theorem, while we postpone for Chapter 8 the most important applications in robust control. In Section 7.6 we give a couple of results on Riccati differential inequalities. Finally, in Section 7.8 we present an alternative approach to the Hermitian Riccati differential equation in terms of Hamiltonian descriptor operators, which parallels the matrix pencil approach to the algebraic Riccati equation.

Chapter 8 contains two important applications of the differential Riccati theory to robust control: the Nehari four-block problem for which we provide the class of all solutions in the suboptimal case, and the disturbance attenuation problem for which we provide solutions in the so-called disturbance feed-forward case.

Chapter 9 is the 'unifying' chapter of the book. The technical machinery employed to investigate the existence of stabilizing solutions for algebraic non­symmetric Riccati equations is similar to that in Chapter 7, involving operator theoretic methods. On the other hand, applications concern Nash and Stackelberg differential games on the infinite time horizon and are directly related to Chapters 6,3 and 2.

An appendix providing basic facts on control theory, Frechet derivatives, and on the implicit function theorem concludes the book.

There is no claim that the subject is completely covered, or that the list of references at the end is exhaustive. The topics presented in detail in this book have been selected according to personal tastes and interests of the authors. Primarily we have focused on those parts of the theory that point to applications in linear control theory and dynamic games. We are aware of the fact that we did not cover important topics related with matrix Riccati equations like:

• Numerical solutions of Riccati equations (see [Mehr91], [Sima96], [IOW97]).

• Polynomial approach to algebraic Riccati equations (see [Fuhr85]).

• Spectral factorization (see [BGK79]' [Fuhr89]' [IOW97], [OaVaOO]), [KaLeOla].

• Superposition formulae (see [AHW82], [SoWi85], [BuWi93], [Egor93], [PadA97], [Wint93]).

• Complex Riccati equations as flows on Cartan-Siegel homogeneity domains (see [Zeli98]).

• Parts of the basic theory of Hermitian algebraic Riccati equations (see [LaRo95], [IOW99]).

• Disconjugacy problems for linear differential and difference equations (see [Copp71], [Krat95a], [Krat95b]' [AhPe96], [Bohn96], [Dosl98]).

Preface xv

This book is a result of a long cooperation between the authors. We deeply regret that V. Ionescu is not anymore among us to see it published. He wrote a first version of his contribution which was nearly finished when he passed away quite unexpectedly in May 2000. We lost a dear friend and esteemed colleague. Fortunately two of his former students, R. ~tefan and C. Oara, who are now both professors at the Polytechnic University of Bucharest, completed and improved considerably the contribution of V. Ionescu. This book is also their work and without their help it would not have been published in its actual form.

Many of our colleagues and students were kind enough to read some parts of the book and to make constructive remarks pointing out new results or suggesting modifications. Particularly we are indebted to A. Hochhaus, who made essential contributions to Sections 2.5, 4.3, 6.7, 6.8, and to D. Kremer who contributed essentially to Chapter 9.

Its a pleasure to record also our sincere thanks to Mrs. Tackenberg and Mrs. Volkmann, who prepared a major part of the Jb.1EX master copy of this book. We would like also to acknowledge the constant support of our institutions: Ecole Normale Superieure de Cachan, University of Duisburg, RWTH-Aachen and Poly­technic University of Bucharest.

Notation

Abbreviations

CALE - the continuous-time algebraic Lyapunov equation

DALE - the discrete-time algebraic Lyapunov equation

RDE - the Riccati differential equation

PRDE - the Periodic Riccati differential equation

HRDE - the Hermitian Riccati differential equation

HRDE(~) - the Hermitian Riccati differential equation associated with the Popov triplet ~

HRDS(~) - the Hermitian Riccati differential system associated with the Popov triplet ~

DRDE - (discrete-time) Riccati difference equation

ARE - algebraic Riccati equation

DARE - discrete-time algebraic Riccati equation

HARE - Hermitian algebraic Riccati equation

HDARE - Hermitian discrete-time algebraic Riccati equation

CARE - HARE with Q = C*C, S = BB*

DUAL - dual algebraic Riccati equation

KPYS - the Kalman-Popov-Yakubovich system

KPYS(~, J) - the Kalman-Popov-Yakubovich system in J-form associated with the Popov triplet ~

HS(~) - the continuous-time (extended) Hamiltonian system associated with the Popov triplet ~

HDO(~) - the Hamiltonian descriptor operator associated with the Popov triplet ~

Notation

ARI - algebraic Riccati inequality

ED - exponentially dichotomic

ES - exponentially stable

EAS - exponentially anti-stable

LARE - the set of all solutions of ARE

LHARE - the set of all solutions of HARE

({HARE - the set of all Hermitian solutions of HARE

PHARE - the set of all positive semidefinite solutions of HARE

PCARE - the set of all positive semidefinite solutions of CARE

NCARE - the set of all negative semidefinite solutions of CARE

Xs - the stabilizing solution of HARE or CARE

Xa - the anti-stabilizing solution of HARE or CARE

X+ - the maximal solution of HARE or CARE

X_ - the minimal solution of HARE or CARE

Numbers, vector spaces, manifolds

IR - the set of real numbers

e - the set of complex numbers

~ (,X) - the real part of ,X E e 'S (,X) - the imaginary part of ,X E e IRmxn - the real vector space of m x n real matrices

em x n - the complex vector space of m x n complex matrices

IRnx1 , enX1 - abbreviated respectively, by IRn, en

L2,n(lR) - the Hilbert space of square integrable functions from IR to en

xvii

RH~xn - the set of all m x n proper, rational matrix valued complex functions, without poles in the closed right part of the complex plane

Co = {z Eel ~ zOO} for 0 E {<,::;, >, 2} - the open or closed half-planes of e e= = {z Eel ~z = O} =: ilR - the imaginary axis

lJ)) = {z E e Ilzl < I} - unit disc

XVlll Notation

j[))o = {Z E C IlzlDl} for 0 E {<,:S;, >, 2} - the open or closed unit disc and its complement

j[))= = {z E c Ilzl = I} = oj[)) - the unit circle

H(n) - the set of all Hermitian n x n - matrices

cn(cm ) - the GraBmann manifold of all n-dimensional subspaces of Cm

£'.(n) - the Lagrange GraBmann (sub-)manifold (of cn(cm ))

el, ... ,en - the standard basis in IRn or cn; the n columns of the n x n identity matrix

[xjx - the coordinate vector of x E IRn or Cn relative to a basis X.

span {Xl, ... ,Xr } - the subspace spanned by Xl, ... ,Xr

M-i-N - direct sum of subspaces

M ED N - orthogonal direct sum of subspaces

M..L - the orthogonal complement to a subspace M

M[..L] = {x E IRn or en I [x, yj = 0 for all y E M} - the orthogonal companion of a subspace with respect to the indefinite scalar product [.,.j in IRn or en

CA,B - the controllable subspace of the pair of matrices (A, B)

UC,A - the unobservable subspace of the pair of matrices (C, A)

Matrices and operators

All matrices are assumed to have real or complex entries. When convenient, an m x n matrix A is also understood as the linear transformation IRn ---t IRm (or en ---t em, as the case may be) represented by A in the standard bases in IRm (or em) and IRn (or en).

In, 1- n X n identity matrix

On, 0 - n x n zero matrix

a(A) - the spectrum of A; the set of distinct eigenvalues of matrix A

A - the complex conjugate (entrywise) of a matrix A

AT - the transpose of A

A * - the conjugate transpose of A

(A*)-l = (A-I)* - is sometimes abbreviated A-*

Al*l - the adjoint of matrix A in the scalar product [.,.j

IIAII = max{ a I a is an eigenvalue of (A* A)I/2} - the operator norm of A induced by the Euclidean norm on en, also called the spectral norm of A.

p(A) = max{I).11 ). is an eigenvalue of A}

Notation xix

diag(A I , ... , Ap) or Al EEl ... EEl Ap - the block diagonal matrix with the diagonal blocks AI, ... , Ap

A ® B - Kronecker product of A and B

A - vec of A

KerA = {x E JRn or Cn I Ax = O} - the kernel, or null-space, of an m x n matrix A

1m A = {Ax I x E JRn or Cn} - the image, or range, of an m x n matrix A

G(W) = 1m ( {v ) - Graph of W E cmxn

Jr (>\) - the r x r upper triangular Jordan block with eigenvalue A

hk (A±ip) - the 2k x 2k almost upper triangular real Jordan block with eigenvalues A ± ip (A E JR, p E JR, wi 0)

For AD E a(A), where A is an n x n matrix, the algebraic multiplicity of AD is dim Ker(A - Ao!)n; the geometric multiplicity of AD is dimKer(A - AD!); the partial multiplicities of AD are the sizes of the Jordan blocks with the eigenvalue AD in the Jordan form of A (these definitions apply also for real matrices A and their non-real eigenvalues AD; then A - AoI is understood as a linear transformation from Cn into Cn ).

R>.o (A) = Ker(A - AoI)n - the root subspace of an n x n matrix A corresponding to AD E a(A) R(A, n) = Rn(A) - the sum of root subspaces of A corresponding to its eigenval­ues in the set n Ro(A) = R(A,Co ) for 0 E {<,::;,=,2:,>} - the sum of root subspaces of A corresponding to its eigenvalues in Co

Rd,O(A) = R(A,[J)o) for 0 E {<,::;,=,>,2:} - the sum of root subspaces of A corresponding to its eigenvalues in [J)o

AIN - the restriction of A to its invariant subspace N Inv(A) - the set of all invariant subspaces of A

<I> A(t, r) - the state transition matrix of the system x(t) = A(t)x(t)

"IJt A (w) = <I> A (w, 0) - monodromy matrix of the w-periodic matrix A

J = (~ -~n) E c2nx2n - the imaginary unit

Jpq = (-Ip 0) E C(p+q)x(p+q) - the signature (or sign) matrix operator o Iq

xx Notation

A > 0 (A ;::: 0) - means that the Hermitian matrix A is positive definite (positive semidefinite)

A > B (A ;::: B) - means that (for Hermitian matrices A and B) A - B is positive definite (positive semidefinite)

In(A) = Crr(A) , v(A), 8(A)) - denotes the number of the eigenvalues of A (counting multiplicities) in C>, C< and C= and it is called the inertia of A.

Ind(A) = (1l"d(A) , vd(A), 8d(A)) - denotes the number of the eigenvalues of A (counting multiplicities) in JD», JD) < and JD)= and it is called the discrete inertia of A

A is called stable (or a Hurwitz matrix) iff a(A) c C<

A is called d-stable (or a Schur matrix) iff a(A) c JD)<

(x, y) - the standard scalar product in IRn or cn or L2,n(lR)

[x, y] - an indefinite scalar product on cn or IRn

lixll = (x, X)I/2 - the Euclidean norm (if not specified otherwise)

liul12 - the L 2 ,m(lR) norm

For H = (_~ ~i*) we define the Riccati operator R(X, H) by

R(X, H) = -A*X -XA-Q+XSX.

T* - the adjoint of the operator T

IITII - the operator induced norm of T

Q~ - the orthogonal projection from L 2 .n(lR) onto L 2,n( -00, T]

P;: - the orthogonal projection from L2,n(lR) onto L2,n[T, 00)

V - the differential operator, (Vx)(t) = dx(t)/dt

n - the time-reversal operator; (nf)(t) = f( -t) RE - the Popov operator associated with the Popov triplet ~

DE(X) - the dissipation matrix operator associated with the Popov triplet ~ and X

[ ~ I ~ ] - a compact notation for the system

± = A(t)x + B(t)u y = C(t)x + D(t)u

[ V- A I-B ] . . C D := C(V- A)-IB + D - the L2 mput-output operator assocIated

with the ED system {A,B,C,D}