Unsolved Problems In Mathematical Systems And Control Theoryedited By Vincent D. Blondel & Alexandre Megretski

Solutions and Partial Solutions

PART 1. LINEAR SYSTEMS 1
Problem 1.1. Stability and composition of transfer functions
      Guillermo Fernández-Anaya, Juan Carlos Martínez-García 3
A Counterexample to Fernández-Martínez's Conjecture, submitted by Wensheng Yu and Long Wang
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
Partial Solution to Problem 3.1 (question 3) submitted by Alex Bombrun and Jean-Baptiste Pomet
Technical details developed further by Alex Bombrun (thesis in French)
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
Partial Solution to Problem 4.1, submitted by Michael Margaliot and João P. Hespanha
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
Partial solution to Problem 6.4, submitted by Daniel Liberzon and Michael Margaliot
Partial solution to Problem 6.4, submitted by Yoav Sharon and Michael Margaliot
Problem 6.5. Robust stability test for interval fractional order linear systems
      Ivo Petrás, YangQuan Chen, Blas M. Vinagre 208
Problem 6.6. Delay-independent and delay-dependent Aizerman problem
      Vladimir Rasvan 212
Partial solution to Problem 6.6, submitted by Dmitry A. Altshuller
Continuation of partial solution to Problem 6.6, submitted by Dmitry A. Altshuller
Continued work on Problem 6.6 and Some Results on Systems with Periodic Right-Hand Sides, submitted by Dmitry A. Altshuller
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
Solution to Problem 10.3, submitted by Paul C. Bell and Igor Potapov
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 µ analysis
      Onur Toker 330

How to submit a solution, partial solution, or comment:

If you would like to submit a solution, partial solution, or comment about one of the problems, please send your comments in a pdf file to Roland Hildebrand (roland.hildebrand@imag.fr) In the subject line of your message, please include:

Blondel: Problem 4.1 (or whatever problem you choose).

All submissions will be considered and all solutions/partial solutions will be reviewed and verified prior to posting.

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