Unsolved Problems in Mathematical Systems and Control Theory

by Vincent D. Blondel, Alexandre Megretski

Overview

This book provides clear presentations of more than sixty important unsolved problems in mathematical systems and control theory. Each of the problems included here is proposed by a leading expert and set forth in an accessible manner. Covering a wide range of areas, the book will be an ideal reference for anyone interested in the latest developments in the field, including specialists in applied mathematics, engineering, and computer science.


The book consists of ten parts representing various problem areas, and each chapter sets forth a different problem presented by a researcher in the particular area and in the same way: description of the problem, motivation and history, available results, and bibliography. It aims not only to encourage work on the included problems but also to suggest new ones and generate fresh research. The reader will be able to submit solutions for possible inclusion on an online version of the book to be updated quarterly on the Princeton University Press website, and thus also be able to access solutions, updated information, and partial solutions as they are developed.

Product Details

ISBN-13:	9780691117485
Publisher:	Princeton University Press
Publication date:	07/26/2004
Pages:	352
Product dimensions:	6.00(w) x 9.25(h) x (d)
Age Range:	18 Years

About the Author

Vincent D. Blondel is Professor of Applied Mathematics and Head of the Department of Mathematical Engineering at the University of Louvain, Louvain-la-Neuve, Belgium. Alexandre Megretski is Associate Professor of Electrical Engineering at Massachusetts Institute of Technology.

Table of Contents

Frontmatter
Preface xiii
Associate Editors xv
Website xvii



PART 1. LINEAR SYSTEMS 1
Problem 1.1. Stability and composition of transfer functions
Guillermo Fernández-Anaya, Juan Carlos Martínez-García 3
Problem 1.2. The realization problem for Herglotz-Nevanlinna functions
Seppo Hassi, Henk de Snoo, Eduard Tsekanovskii 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 Martínez-García, Michel Malabre, Vladimir Kucera 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
Problem 1.9. A Farkas lemma for behavioral inequalities
A.A. (Tonny) ten Dam, J.W. (Hans) Nieuwenhuis 40
Problem 1.10. Regular feedback implementability of linear differential behaviors
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
Problem 2.1. On error of estimation and minimum of cost for wide band noise 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
Problem 2.4. On the convergence of normal forms for analytic control systems
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
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
Problem 3.6. Does cheap control solve a singular nonlinear quadratic problem?
Yuri V. Orlov 111
Problem 3.7. Delta-Sigma modulator synthesis
Anders Rantzer 114
Problem 3.8. Determining of various asymptotics of solutions of nonlinear time-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. L₂-induced gains of switched linear systems
João 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
Problem 5.2. The dynamical Lame system with boundary control: on the structure of reachable sets
M.I. Belishev 160
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. Çamlibel, 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 Petrs, YangQuan Chen, Blas M. Vinagre 208
Problem 6.6. Delay-independent and delay-dependent Aizerman problem
Vladimir Rasvan 212
Problem 6.7. Open problems in control of linear discrete multidimensional systems
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
Problem 6.9. Generalized Lyapunov theory and its omega-transformable regions
Sheng-Guo Wang 233
Problem 6.10. Smooth Lyapunov characterization of measurement to error stability
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_[infinity]-norm approximation
A.C. Antoulas, A. Astolfi 267
Problem 8.2. Noniterative computation of optimal value in H_[infinity] control
Ben M. Chen 271
Problem 8.3. Determining the least upper bound on the achievable delay margin
Daniel E. Davison, Daniel E. Miller 276
Problem 8.4. Stable controller coefficient perturbation in floating point implementation
Jun Wu, Sheng Chen 280



PART 9. IDENTIFICATION, SIGNAL PROCESSING 285
Problem 9.1. A conjecture on Lyapunov equations and principal angles in sub-space identification
Katrien De Cock, Bart De Moor 287
Problem 9.2. Stability of a nonlinear adaptive system for filtering and parameter estimation
Masoud Karimi-Ghartemani, Alireza K. Ziarani 293



PART 10. ALGORITHMS, COMPUTATION 297
Problem 10.1. Root-clustering for multivariate polynomials and robust stability analysis
Pierre-Alexandre Bliman 299
Problem 10.2. When is a pair of matrices stable?
Vincent D. Blondel, Jacques Theys, John N. Tsitsiklis 304
Problem 10.3. Freeness of multiplicative matrix semigroups
Vincent D. Blondel, Julien Cassaigne, Juhani Karhumäki 309
Problem 10.4. Vector-valued quadratic forms in control theory
Francesco Bullo, Jorge Cortés, Andrew D. Lewis, Sonia Martínez 315
Problem 10.5. Nilpotent bases of distributions
Henry G. Hermes, Matthias Kawski 321
Problem 10.6. What is the characteristic polynomial of a signal flow graph?
Andrew D. Lewis 326
Problem 10.7. Open problems in randomized [mu] analysis
Onur Toker 330

What People are Saying About This

Hager

This is an extremely important book that presents, in a clear way, many important and stimulating mathematical problems in systems and control. It will be an important reference for both researchers and people outside the field.
— William W. Hager, University of Florida

Kemin Zhou

This book covers a wide range of systems from linear to nonlinear, deterministic to stochastic, finite dimensional to infinite dimensional, and so on. It includes at least some set of problems that will interest any researcher in the field.
— Kemin Zhou, Louisiana State University

From the Publisher

"This is an extremely important book that presents, in a clear way, many important and stimulating mathematical problems in systems and control. It will be an important reference for both researchers and people outside the field."—William W. Hager, University of Florida

"This book covers a wide range of systems from linear to nonlinear, deterministic to stochastic, finite dimensional to infinite dimensional, and so on. It includes at least some set of problems that will interest any researcher in the field."—Kemin Zhou, Louisiana State University

Recipe

"This is an extremely important book that presents, in a clear way, many important and stimulating mathematical problems in systems and control. It will be an important reference for both researchers and people outside the field."—William W. Hager, University of Florida

"This book covers a wide range of systems from linear to nonlinear, deterministic to stochastic, finite dimensional to infinite dimensional, and so on. It includes at least some set of problems that will interest any researcher in the field."—Kemin Zhou, Louisiana State University

Editorial Reviews

"This is an extremely important book that presents, in a clear way, many important and stimulating mathematical problems in systems and control. It will be an important reference for both researchers and people outside the field."—William W. Hager, University of Florida


"This book covers a wide range of systems from linear to nonlinear, deterministic to stochastic, finite dimensional to infinite dimensional, and so on. It includes at least some set of problems that will interest any researcher in the field."—Kemin Zhou, Louisiana State University

From the Publisher