Advances in Mathematical Systems Theory

This volume contains the lectures presented at the workshop on "Advances in Mathematical Systems Theory," held on the island of Borkum, Germany (April 20-23, 1999). The book will be of interest to graduate students and researchers interested in control theory and mathematical systems theory, who will find in-depth analysis and presentations from diverse perspectives interacting in this lively area. The editors are proud to dedicate this volume to Diederich Hinrichsen on the occasion of his 60th birthday in acknowl edgment of his major contributions to linear systems theory and control theory and his long-term achievements in establishing mathematical sys tems theory in Germany. We all owe much to him as a teacher, colleague, and friend. The editors thank the Graduiertenkolleg "Komplexe Dynamische Sys teme" at the University of Bremen as well as the European "Nonlinear Control Network" for providing financial support that enabled this work shop. Augsburg, Germany Fritz Colonius Wiirzburg, Germany Uwe Helmke Kaiserslautern, Germany Dieter Pratzel-Wolters Bremen, Germany Fabian Wirth Introduction The workshop "Advances in Mathematical Systems Theory" took place in honor of Diederich Hinrichsen on the occasion of his 60th birthday. The following chapters are based on invited lectures and cover a wide range of topics in linear and nonlinear systems theory including parameteriza tion problems, behaviors of linear systems and convolutional codes, as well as complementarity systems and hybrid systems.

This self-contained volume surveys advances in mathematical systemstheory and control. Fifteen chapters by leading researchers addresscross-section of major research directions: hybrid systems theoryrobust control and stability of linear and nonlinear systemsparametrization of linear systems and control of infinite dimensionalsystems.Professionals and researchers in systems and control engineeringapplied mathematics, and systems engineering will find the book auseful resource.

1 Transitory Behavior of Uncertain Systems.- 1.1 Introduction.- 1.2 Transient Excursions.- 1.3 Pseudospectra and Spectral Value Sets.- 1.4 State Feedback.- 1.5 Transient Excursions of Uncertain Systems.- 2 Robust Stability of Multivariate Polynomials.- 2.1 Introduction.- 2.2 Basic Notions and Definitions.- 2.2.1 Classes of Stable Polynomials.- 2.3 Properties of Stable Polynomials.- 2.4 Zero Exclusion Principle.- 3 Robustness of Nonlinear Systems and Their Domains of Attraction.- 3.1 Introduction.- 3.2 Preliminaries.- 3.3 Linearization Theory.- 3.4 Calculating the Local Stability Radius.- 3.5 Robust Domains of Attraction.- 3.6 An Optimal Control Characterization of the Robust Domain of Attraction.- 3.7 Conclusion.- 4 On Stability Radii of Slowly Time-Varying Systems.- 4.1 Introduction.- 4.2 Perturbation Classes.- 4.3 Stability of Nonlinearly Perturbed Linear Systems.- 4.4 Stability Radii of Slowly Time-Varying Systems.- 4.5 Two Lemmas.- 4.6 Conclusions.- 5 An Invariance Radius for Nonlinear Systems.- 5.1 Introduction.- 5.2 Background on Invariant Control Sets and Chain Control Sets.- 5.3 An Invariance Radius for Nonlinear Systems.- 6 State and Continuity.- 6.1 Introduction.- 6.2 Linear Differential Systems.- 6.3 Latent Variables.- 6.4 State Representations.- 6.5 Smoothing Functionals.- 6.6 Main Results.- 6.7 Proofs.- 6.8 Remarks.- 7 Parameterization of Conditioned Invariant Subspaces.- 7.1 Introduction.- 7.2 Preliminaries.- 7.3 On Conditioned Invariant Subspaces.- 7.4 The State Space Approach.- 7.5 On the Parameterization of Conditioned Invariant Subspaces.- 7.6 Topology of Tight Conditioned Invariant Subspaces.- 7.7 Brunovsky Strata for Conditioned Invariant Subspaces...- 8 Duality Between Multidimensional Convolutional Codes and Systems.- 8.1 Introduction.- 8.2 Multidimensional Convolutional Codes.- 8.3 Duality Between Codes and Behaviors.- 8.4 First-Order Representations for One-Dimensional Codes.- 8.5 Conclusion.- 9 Control of Rate-Bounded Hybrid Systems with Liveness Constraints.- 9.1 Introduction.- 9.2 Hybrid Machines.- 9.3 Liveness.- 9.4 Control.- 10 A General Principle of Marked Extraction.- 10.1 Introduction.- 10.2 Joint Production and the Substitution Theorem.- 10.3 Marked Extraction in Convex Sets.- 10.4 The Factorial Complex of a Krull Monoid.- 11 Between Mathematical Programming and Systems Theory: Linear Complementarity Systems.- 11.1 Introduction.- 11.2 Examples.- 11.3 Existence and uniqueness of solutions.- 11.4 Linear complementarity systems.- 11.5 A distributional interpretation.- 11.6 Well-posedness.- 11.7 Relay systems.- 11.8 Discontinuous dependence on initial conditions.- 11.9 Conclusions.- 12 Exact Controllability of Co-groups with One-Dimensional Input Operators.- 12.1 Introduction.- 12.2 System Description.- 12.3 Exact Controllability.- 12.4 Proofs of the Main Results.- 12.5 An Example.- 13 Normalized Coprime Factorizations for Strongly Stabilizable Systems.- 13.1 Introduction.- 13.2 Problem Formulation and Mathematical Background.- 13.3 Formulae for Normalized Coprime Factorizations.- 13.4 Application to Hybrid Flexible Structures.- 14 Low-Gain Integral Control of Infinite-Dimensional Regular Linear Systems Subject to Input Hysteresis.- 14.1 Introduction.- 14.2 Preliminaries on Regular Linear Systems.- 14.3 A Class of Causal Monotone Nonlinear Operators.- 14.4 Integral Control in the Presence of Input Nonlinearities Satisfying (N1) to (N8).- 14.5 Hysteresis Nonlinearities Satisfying (N1) to (N8).- 14.6 Example: Controlled Diffusion Process with Output Delay.- 14.7 Appendix.