Unsolved Problems in Mathematical Systems and Control Theory

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.

"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

• Frontmatter Preface xiii Associate Editors xv Website xvii PART 1. LINEAR SYSTEMS 1 Problem 1.1. Stability and composition of transfer functions Guillermo Fernandez-Anaya, Juan Carlos Martinez-Garcia 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 Martinez-Garcia, 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 Joao 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. Camlibel, 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 Karhumaki 309 Problem 10.4. Vector-valued quadratic forms in control theory Francesco Bullo, Jorge Cortes, Andrew D. Lewis, Sonia Martinez 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