**Unsolved Problems in**

Mathematical Systems and

Control Theory

Mathematical Systems and

Control Theory

Edited by

Vincent D. Blondel

Alexandre Megretski

PRINCETON UNIVERSITY PRESS

PRINCETON AND OXFORD

-iii-

Mathematical Systems and

Control Theory

Edited by

Vincent D. Blondel

Alexandre Megretski

PRINCETON UNIVERSITY PRESS

PRINCETON AND OXFORD

-iii-

Questia, a part of Gale, Cengage Learning. www.questia.com

**Publication information:**
Book title: Unsolved Problems in Mathematical Systems and Control Theory.
Contributors: Vincent D. Blondel - Editor, Alexandre Megretski - Editor.
Publisher: Princeton University Press.
Place of publication: Princeton, NJ.
Publication year: 2004.
Page number: iii.

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Unsolved Problems in Mathematical Systems and Control Theory

Table of contents
Table of contents
#### Table of contents

- Title Page iii
- Contents vii
- Preface xiii
- Associate Editors xv
- Website xvii
- Part 1 - Linear Systems 1
- Problem 1.1 - Stability and Composition of Transfer Functions 3
- Problem 1.2 - The Realization Problem for Herglotz-Nevanlinna Functions 8
- Problem 1.3 - Does Any Analytic Contractive Operator Function on the Polydisk Have a Dissipative Scattering ND Realization? 14
- Problem 1.4 - Partial Disturbance Decoupling with Stability 18
- Problem 1.5 - Is Monopoli's Model Reference Adaptive Controller Correct? 22
- Problem 1.6 - Model Reduction of Delay Systems 29
- Problem 1.7 - Schur Extremal Problems 33
- Problem 1.8 - The Elusive Iff Test for Time-Controllability of Behaviours 36
- Problem 1.9 - A Farkas Lemma for Behavioral Inequalities 40
- Problem 1.10 - Regular Feedback Implementability of Linear Differential Behaviors 44
- Problem 1.11 - Riccati Stability 49
- Problem 1.12 - State and First Order Representations 54
- Problem 1.13 - Projection of State Space Realizations 58
- Part 2 - Stochastic Systems 65
- Problem 2.1 - On Error of Estimation and Minimum of Cost for Wide Band Noise Driven Systems 67
- Problem 2.2 - On the Stability of Random Matrices 71
- Problem 2.3 - Aspects of Fisher Geometry for Stochastic Linear Systems 76
- Problem 2.4 - On the Convergence of Normal Forms for Analytic Control Systems 82
- Part 3 - Nonlinear Systems 87
- Problem 3.1 - Minimum Time Control of the Kepler Equation 89
- Problem 3.2 - Linearization of Linearly Controllable Systems 93
- Problem 3.3 - Bases for Lie Algebras and a Continuous Cbh Formula 97
- Problem 3.4 - An Extended Gradient Conjecture 103
- Problem 3.5 - Optimal Transaction Costs from a Stackelberg Perspective 107
- Problem 3.6 - Does Cheap Control Solve a Singular Nonlinear Quadratic Problem? 111
- Problem 3.7 - Delta-Sigma Modulator Synthesis 114
- Problem 3.8 - Determining of Various Asymptotics of Solutions of Nonlinear Time-Optimal Problems Via Right Ideals in the Moment Algebra 117
- Problem 3.9 - Dynamics of Principal and Minor Component Flows 122
- Part 4 - Discrete Event, Hybrid Systems 129
- Problem 4.1 - L2-Induced Gains of Switched Linear Systems 131
- Problem 4.2 - The State Partitioning Problem of Quantised Systems 134
- Problem 4.3 - Feedback Control in Flowshops 140
- Problem 4.4 - Decentralized Control with Communication between Controllers 144
- Part 5 - Distributed Parameter Systems 151
- Problem 5.1 - Infinite Dimensional Backstepping for Nonlinear Parabolic Pdes 153
- Problem 5.2 - The Dynamical Lame System with Boundary Control: on the Structure of Reachable Sets 160
- Problem 5.3 - Null-Controllability of the Heat Equation in Unbounded Domains 163
- Problem 5.4 - Is the Conservative Wave Equation Regular? 169
- Problem 5.5 - Exact Controllability of the Semi-Linear Wave Equation 173
- Problem 5.6 - Some Control Problems in Electromagnetics and Fluid Dynamics 179
- Part 6 - Stability, Stabilization 187
- Problem 6.1 - Copositive Lyapunov Functions 189
- Problem 6.2 - The Strong Stabilization Problem for Linear Time-Varying Systems 194
- Problem 6.3 - Robustness of Transient Behavior 197
- Problem 6.4 - Lie Algebras and Stability of Switched Nonlinear Systems 203
- Problem 6.5 - Robust Stability Test for Interval Fractional Order Linear Systems 208
- Problem 6.6 - Delay Independent and Delay Dependent Aizerman Problem 212
- Problem 6.7 - Open Problems in Control of Linear Discrete Multidimensional Systems 221
- Problem 6.8 - An Open Problem in Adaptative Nonlinear Control Theory 229
- Problem 6.9 - Generalized Lyapunov Theory and Its Omega-Transformable Regions 233
- Problem 6.10 - Smooth Lyapunov Characterization of Measurement to Error Stability 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 247
- Problem 7.2 - A Hautus Test for Infinite-Dimensional Systems 251
- Problem 7.3 - Three Problems in the Field of Observability 256
- Problem 7.4 - Control of the Kdv Equation 260
- Part 8 - Robustness, Robust Control 265
- Problem 8.1 - H∞-Norm Approximation 267
- Problem 8.2 - Non-Iterative Computation of Optimal Value in H∞ Control 271
- Problem 8.3 - Determining the Least Upper Bound on the Achievable Delay Margin 276
- Problem 8.4 - Stable Controller Coefficient Perturbation in Floating Point Implementation 280
- Part 9 - Identification, Signal Processing 285
- Problem 9.1 - A Conjecture on Lyapunov Equations and Principal 287
- Problem 9.2 - Stability of a Nonlinear Adaptive System for Filtering and Parameter Estimation 293
- Part 10 - Algorithms, Computation 297
- Problem 10.1 - Root-Clustering for Multivariate Polynomials and Robust Stability Analysis 299
- Problem 10.2 - When Is a Pair of Matrices Stable? 304
- Problem 10.3 - Freeness of Multiplicative Matrix Semigroups 309
- Problem 10.4 - Vector-Valued Quadratic Forms in Control Theory 315
- Problem 10.5 - Nilpotent Bases of Distributions 321
- Problem 10.6 - What Is the Characteristic Polynomial of a Signal Flow Graph? 326
- Problem 10.7 - Open Problems in Randomized µ Analysis 330

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