Unsolved Problems in Mathematical Systems and Control Theory

351 Pages·2004·1.456 MB·English

by  Vincent D. Blondel, Alexandre Megretski

Unsolved Problems in
Mathematical Systems and
Control Theory
Edited by
Vincent D. Blondel
Alexandre Megretski
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
iv
Copyright (cid:13)c 2004 by Princeton University Press
PublishedbyPrincetonUniversityPress,41WilliamStreet,Princeton,New
Jersey 08540, USA
IntheUnitedKingdom: PrincetonUniversityPress,3MarketPlace,Wood-
stock, Oxfordshire OX20 1SY, UK
All rights reserved
Library of Congress Cataloging-in-Publication Data
Unsolved problems in mathematical systems and control theory
Edited by Vincent D. Blondel, Alexandre Megretski. p. cm.
Includes bibliographical references.
ISBN 0-691-11748-9 (cl : alk. paper)
1. System analysis. 2. Control theory. I. Blondel, Vincent. II. Megretski,
Alexandre.
QA402.U535 2004 2003064802
003—dc22
The publisher would like to acknowledge the editors of this volume for pro-
viding the camera-ready copy from which this book was printed.
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
I have yet to see any problem, however complicated, which, when
you looked at it in the right way, did not become still more compli-
cated.
Poul Anderson

Contents
Preface xiii
Associate Editors xv
Website xvii
PART 1. LINEAR SYSTEMS 1
Problem 1.1. Stability and composition of transfer functions
Guillermo Fern´andez-Anaya, Juan Carlos Mart´ınez-Garc´ıa 3
Problem 1.2. The realization problem for Herglotz-Nevanlinna functions
Seppo Hassi, Henk de Snoo, Eduard Tsekanovski˘ı 8
Problem 1.3. Does any analytic contractive operator function on the polydisk
have a dissipative scattering nD realization?
Dmitry S. Kalyuzhniy-Verbovetzky 14
Problem 1.4. Partial disturbance decoupling with stability
Juan Carlos Mart´ınez-Garc´ıa, Michel Malabre, Vladimir Kuˇcera 18
Problem 1.5. Is Monopoli’s model reference adaptive controller correct?
A. S. Morse 22
Problem 1.6. Model reduction of delay systems
Jonathan R. Partington 29
Problem 1.7. Schur extremal problems
Lev Sakhnovich 33
Problem 1.8. The elusive iff test for time-controllability of behaviors
Amol J. Sasane 36
viii CONTENTS
Problem 1.9. A Farkas lemma for behavioral inequalities
A.A. (Tonny) ten Dam, J.W. (Hans) Nieuwenhuis 40
Problem1.10. Regularfeedbackimplementabilityoflineardifferentialbehaviors
H. L. Trentelman 44
Problem 1.11. Riccati stability
Erik I. Verriest 49
Problem 1.12. State and first order representations
Jan C. Willems 54
Problem 1.13. Projection of state space realizations
Antoine Vandendorpe, Paul Van Dooren 58
PART 2. STOCHASTIC SYSTEMS 65
Problem2.1. Onerrorofestimationandminimumofcostforwidebandnoise
driven systems
Agamirza E. Bashirov 67
Problem 2.2. On the stability of random matrices
Giuseppe C. Calafiore, Fabrizio Dabbene 71
Problem 2.3. Aspects of Fisher geometry for stochastic linear systems
Bernard Hanzon, Ralf Peeters 76
Problem2.4. Ontheconvergenceofnormalformsforanalyticcontrolsystems
Wei Kang, Arthur J. Krener 82
PART 3. NONLINEAR SYSTEMS 87
Problem 3.1. Minimum time control of the Kepler equation
Jean-Baptiste Caillau, Joseph Gergaud, Joseph Noailles 89
Problem 3.2. Linearization of linearly controllable systems
R. Devanathan 93
Problem 3.3. Bases for Lie algebras and a continuous CBH formula
Matthias Kawski 97
CONTENTS ix
Problem 3.4. An extended gradient conjecture
Luis Carlos Martins Jr., Geraldo Nunes Silva 103
Problem 3.5. Optimal transaction costs from a Stackelberg perspective
Geert Jan Olsder 107
Problem3.6. Doescheapcontrolsolveasingularnonlinearquadraticproblem?
Yuri V. Orlov 111
Problem 3.7. Delta-Sigma modulator synthesis
Anders Rantzer 114
Problem3.8. Determiningofvariousasymptoticsofsolutionsofnonlineartime-
optimal problems via right ideals in the moment algebra
G. M. Sklyar, S. Yu. Ignatovich 117
Problem 3.9. Dynamics of principal and minor component flows
U. Helmke, S. Yoshizawa, R. Evans, J.H. Manton, and I.M.Y. Mareels 122
PART 4. DISCRETE EVENT, HYBRID SYSTEMS 129
Problem 4.1. L -induced gains of switched linear systems
2
Jo˜ao P. Hespanha 131
Problem 4.2. The state partitioning problem of quantized systems
Jan Lunze 134
Problem 4.3. Feedback control in flowshops
S.P. Sethi and Q. Zhang 140
Problem 4.4. Decentralized control with communication between controllers
Jan H. van Schuppen 144
PART 5. DISTRIBUTED PARAMETER SYSTEMS 151
Problem 5.1. Infinite dimensional backstepping for nonlinear parabolic PDEs
Andras Balogh, Miroslav Krstic 153
Problem5.2. ThedynamicalLamesystemwithboundarycontrol: onthestruc-
ture of reachable sets
M.I. Belishev 160
x CONTENTS
Problem 5.3. Null-controllability of the heat equation in unbounded domains
Sorin Micu, Enrique Zuazua 163
Problem 5.4. Is the conservative wave equation regular?
George Weiss 169
Problem 5.5. Exact controllability of the semilinear wave equation
Xu Zhang, Enrique Zuazua 173
Problem 5.6. Some control problems in electromagnetics and fluid dynamics
Lorella Fatone, Maria Cristina Recchioni, Francesco Zirilli 179
PART 6. STABILITY, STABILIZATION 187
Problem 6.1. Copositive Lyapunov functions
M. K. C¸amlıbel, J. M. Schumacher 189
Problem 6.2. The strong stabilization problem for linear time-varying systems
Avraham Feintuch 194
Problem 6.3. Robustness of transient behavior
Diederich Hinrichsen, Elmar Plischke, Fabian Wirth 197
Problem 6.4. Lie algebras and stability of switched nonlinear systems
Daniel Liberzon 203
Problem 6.5. Robust stability test for interval fractional order linear systems
Ivo Petr´aˇs, YangQuan Chen, Blas M. Vinagre 208
Problem 6.6. Delay-independent and delay-dependent Aizerman problem
Vladimir R˘asvan 212
Problem6.7. Openproblemsincontroloflineardiscretemultidimensionalsys-
tems
Li Xu, Zhiping Lin, Jiang-Qian Ying, Osami Saito, Yoshihisa Anazawa 221
Problem 6.8. An open problem in adaptative nonlinear control theory
Leonid S. Zhiteckij 229
Problem6.9. GeneralizedLyapunovtheoryanditsomega-transformableregions
Sheng-Guo Wang 233
CONTENTS xi
Problem6.10. SmoothLyapunovcharacterizationofmeasurementtoerrorsta-
bility
Brian P. Ingalls, Eduardo D. Sontag 239
PART 7. CONTROLLABILITY, OBSERVABILITY 245
Problem 7.1. Time for local controllability of a 1-D tank containing a fluid
modeled by the shallow water equations
Jean-Michel Coron 247
Problem 7.2. A Hautus test for infinite-dimensional systems
Birgit Jacob, Hans Zwart 251
Problem 7.3. Three problems in the field of observability
Philippe Jouan 256
Problem 7.4. Control of the KdV equation
Lionel Rosier 260
PART 8. ROBUSTNESS, ROBUST CONTROL 265
Problem 8.1. H∞-norm approximation
A.C. Antoulas, A. Astolfi 267
Problem 8.2. Noniterative computation of optimal value in H∞ control
Ben M. Chen 271
Problem8.3. Determiningtheleastupperboundontheachievabledelaymargin
Daniel E. Davison, Daniel E. Miller 276
Problem 8.4. Stable controller coefficient perturbation infloating point imple-
mentation
Jun Wu, Sheng Chen 280
PART 9. IDENTIFICATION, SIGNAL PROCESSING 285
Problem9.1. AconjectureonLyapunovequationsandprincipalanglesinsub-
space identification
Katrien De Cock, Bart De Moor 287