Unsolved Problems in Mathematical Systems and Control Theory
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Unsolved Problems in Mathematical Systems and Control Theory

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Vincent D. Blondel
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FrontmatterPreface xiiiAssociate Editors xvWebsite xvii
PART 1. LINEAR SYSTEMS 1Problem 1.1. Stability and composition of transfer functionsGuillermo Fernández-Anaya, Juan Carlos Martínez-García 3Problem 1.2. The realization problem for Herglotz-Nevanlinna functionsSeppo Hassi, Henk de Snoo, Eduard Tsekanovskii 8Problem 1.3. Does any analytic contractive operator function on the polydisk have a dissipative scattering nD realization?Dmitry S. Kalyuzhniy-Verbovetzky 14Problem 1.4. Partial disturbance decoupling with stabilityJuan Carlos Martínez-García, Michel Malabre, Vladimir Kucera 18Problem 1.5. Is Monopoli's model reference adaptive controller correct?A. S. Morse 22Problem 1.6. Model reduction of delay systemsJonathan R. Partington 29Problem 1.7. Schur extremal problemsLev Sakhnovich 33Problem 1.8. The elusive iff test for time-controllability of behaviorsAmol J. Sasane 36Problem 1.9. A Farkas lemma for behavioral inequalitiesA.A. (Tonny) ten Dam, J.W. (Hans) Nieuwenhuis 40Problem 1.10. Regular feedback implementability of linear differential behaviorsH. L. Trentelman 44Problem 1.11. Riccati stabilityErik I. Verriest 49Problem 1.12. State and first order representationsJan C. Willems 54Problem 1.13. Projection of state space realizationsAntoine Vandendorpe, Paul Van Dooren 58

PART 2. STOCHASTIC SYSTEMS 65Problem 2.1. On error of estimation and minimum of cost for wide band noise driven systemsAgamirza E. Bashirov 67Problem 2.2. On the stability of random matricesGiuseppe C. Calafiore, Fabrizio Dabbene 71Problem 2.3. Aspects of Fisher geometry for stochastic linear systemsBernard Hanzon, Ralf Peeters 76Problem 2.4. On the convergence of normal forms for analytic control systemsWei Kang, Arthur J. Krener 82

PART 3. NONLINEAR SYSTEMS 87Problem 3.1. Minimum time control of the Kepler equationJean-Baptiste Caillau, Joseph Gergaud, Joseph Noailles 89Problem 3.2. Linearization of linearly controllable systemsR. Devanathan 93Problem 3.3. Bases for Lie algebras and a continuous CBH formulaMatthias Kawski 97Problem 3.4. An extended gradient conjectureLuis Carlos Martins Jr., Geraldo Nunes Silva 103Problem 3.5. Optimal transaction costs from a Stackelberg perspectiveGeert Jan Olsder 107Problem 3.6. Does cheap control solve a singular nonlinear quadratic problem?Yuri V. Orlov 111Problem 3.7. Delta-Sigma modulator synthesisAnders Rantzer 114Problem 3.8. Determining of various asymptotics of solutions of nonlinear time-optimal problems via right ideals in the moment algebraG. M. Sklyar, S. Yu. Ignatovich 117Problem 3.9. Dynamics of principal and minor component flowsU. Helmke, S. Yoshizawa, R. Evans, J.H. Manton, and I.M.Y. Mareels 122

PART 4. DISCRETE EVENT, HYBRID SYSTEMS 129Problem 4.1. L2-induced gains of switched linear systemsJoão P. Hespanha 131Problem 4.2. The state partitioning problem of quantized systemsJan Lunze 134Problem 4.3. Feedback control in flowshopsS.P. Sethi and Q. Zhang 140Problem 4.4. Decentralized control with communication between controllersJan H. van Schuppen 144

PART 5. DISTRIBUTED PARAMETER SYSTEMS 151Problem 5.1. Infinite dimensional backstepping for nonlinear parabolic PDEsAndras Balogh, Miroslav Krstic 153Problem 5.2. The dynamical Lame system with boundary control: on the structure of reachable setsM.I. Belishev 160Problem 5.3. Null-controllability of the heat equation in unbounded domainsSorin Micu, Enrique Zuazua 163Problem 5.4. Is the conservative wave equation regular?George Weiss 169Problem 5.5. Exact controllability of the semilinear wave equationXu Zhang, Enrique Zuazua 173Problem 5.6. Some control problems in electromagnetics and fluid dynamicsLorella Fatone, Maria Cristina Recchioni, Francesco Zirilli 179

PART 6. STABILITY, STABILIZATION 187Problem 6.1. Copositive Lyapunov functionsM. K. Çamlibel, J. M. Schumacher 189Problem 6.2. The strong stabilization problem for linear time-varying systemsAvraham Feintuch 194Problem 6.3. Robustness of transient behaviorDiederich Hinrichsen, Elmar Plischke, Fabian Wirth 197Problem 6.4. Lie algebras and stability of switched nonlinear systemsDaniel Liberzon 203Problem 6.5. Robust stability test for interval fractional order linear systemsIvo Petrs, YangQuan Chen, Blas M. Vinagre 208Problem 6.6. Delay-independent and delay-dependent Aizerman problemVladimir Rasvan 212Problem 6.7. Open problems in control of linear discrete multidimensional systemsLi Xu, Zhiping Lin, Jiang-Qian Ying, Osami Saito, Yoshihisa Anazawa 221Problem 6.8. An open problem in adaptative nonlinear control theoryLeonid S. Zhiteckij 229Problem 6.9. Generalized Lyapunov theory and its omega-transformable regionsSheng-Guo Wang 233Problem 6.10. Smooth Lyapunov characterization of measurement to error stabilityBrian P. Ingalls, Eduardo D. Sontag 239

PART 7. CONTROLLABILITY, OBSERVABILITY 245Problem 7.1. Time for local controllability of a 1-D tank containing a fluid modeled by the shallow water equationsJean-Michel Coron 247Problem 7.2. A Hautus test for infinite-dimensional systemsBirgit Jacob, Hans Zwart 251Problem 7.3. Three problems in the field of observabilityPhilippe Jouan 256Problem 7.4. Control of the KdV equationLionel Rosier 260

PART 8. ROBUSTNESS, ROBUST CONTROL 265Problem 8.1. H[infinity]-norm approximationA.C. Antoulas, A. Astolfi 267Problem 8.2. Noniterative computation of optimal value in H[infinity] controlBen M. Chen 271Problem 8.3. Determining the least upper bound on the achievable delay marginDaniel E. Davison, Daniel E. Miller 276Problem 8.4. Stable controller coefficient perturbation in floating point implementationJun Wu, Sheng Chen 280

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

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

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