Advances in Mathematical Systems Theory

This self-contained volume surveys advances in mathematical systems theory and control. Fifteen chapters by leading researchers address cross-section of major research directions: hybrid systems theory, robust control and stability of linear and nonlinear systems, parametrization of linear systems and control of infinite dimensional systems. Professionals and researchers in systems and control engineering, applied mathematics, and systems engineering will find the book a useful resource.

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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.- 2.4.1 Families of Polynomials.- 2.4.2 Zero Exclusion Principle.- 2.4.3 Edge Theorem.- 2.4.4 Stability Radius.- 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.- 11.2.1 Circuits with ideal diodes.- 11.2.2 Mechanical systems with unilateral constraints.- 11.2.3 Optimal control with state constraints.- 11.2.4 Variable-structure systems.- 11.2.5 Piecewise linear systems.- 11.2.6 Projected dynamical systems.- 11.2.7 Diffusion with a free boundary.- 11.2.8 Max-plus systems.- 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.