The Dynamics of Control

1 Introduction.- 2 Dynamics, Perturbations, and Control.- 2.1 Perturbations of Complex behavior.- 2.2 Approximation of Complex Systems.- 2.3 Generic Behavior of Perturbations.- 2.4 Stability Boundaries and Multistability.- 2.5 Reachability in Control Systems.- 2.6 Linear and Nonlinear Stability Radii.- 2.7 Stabilization of Bilinear Systems.- 2.8 The Lyapunov Spectrum of Matrices.- I Global Theory.- 3 Control Sets.- 4 Control Flows and Limit behavior.- II Linearization Theory.- 5 Linear Flows on Vector Bundles.- 6 Bilinear Systems on Vector Bundles.- 7 Linearization at a Singular Point.- III Applications.- 8 One-Dimensional Control Systems.- 9 Examples for Global behavior.- 10 Examples for the Spectrum.- 11 Stability Radii and Robust Stability.- 12 Open and Closed Loop Stabilization.- 13 Dynamics of Perturbations.- IV Appendices.- A Geometric Control Theory.- A.1 Differentiable Manifolds and Vector Fields.- A.2 Basic Definitions for Control Systems.- A.3 The Orbit Theorem.- A.4 Local Accessibility.- A.5 Notes.- B Dynamical Systems.- B.1 Vector Bundles.- B.2 Morse Decompositions, Attractors, Chains.- B.3 Ergodic Theory.- B.4 Notes.- C Numerical Computation of Orbits.- C.1 Orbits and Approximately Invariant Sets.- C.2 Computing Approximately Invariant Sets.- C.3 Computation via Time Optimal Control.- C.4 Notes.- D.1 Problem Formulation and Main Results.- D.2 Discounted and Average Functional.- D.3 Approximation of the Spectrum.- D.4 The Hamilton-Jacobi-Bellman Equation.- D.5 Discounted Optimal Control Problems.- D.6 Notes.

From the reviews:"The scope of the present book is comprehensive and includes a variety of topics related to the theory of systems and control. The main frame is the notion of control flow with its global dynamics and linearization. Time-varying perturbations, global and local dynamics of control systems, as well as relevant numerical methods are treated.... The book is an excellent resource for the foundations and applications of control theory and nonlinear dynamics for mathematicians, physicists, and engineers. The point of view of the authors, who incorporated into this treatise a variety of topics from fields that were often considered rather separately, makes the book especially interesting." -Applications of Mathematics"The Dynamics of Control provides a non traditional approach to the subjects of Dynamical Systems theory and Control theory. It is probably the first book where such notions as Morse decomposition, chain recurrence from dynamical systems theory are systematically used for control systems. ... self contained appendix is very helpful. ... The book contains many examples which are helpful in mastering the material. ... Also, the systematic use of dynamical systems theory as applied to control systems should be interesting to control theorists." (V. Zharnitsky, Dynamical Systems Magazine, April, 2008)

From the reviews:

"The scope of the present book is comprehensive and includes a variety of topics related to the theory of systems and control. The main frame is the notion of control flow with its global dynamics and linearization. Time-varying perturbations, global and local dynamics of control systems, as well as relevant numerical methods are treated.... The book is an excellent resource for the foundations and applications of control theory and nonlinear dynamics for mathematicians, physicists, and engineers. The point of view of the authors, who incorporated into this treatise a variety of topics from fields that were often considered rather separately, makes the book especially interesting." -Applications of Mathematics

"The Dynamics of Control provides a non traditional approach to the subjects of Dynamical Systems theory and Control theory. It is probably the first book where such notions as Morse decomposition, chain recurrence from dynamical systems theory are systematically used for control systems. ... self contained appendix is very helpful. ... The book contains many examples which are helpful in mastering the material. ... Also, the systematic use of dynamical systems theory as applied to control systems should be interesting to control theorists." (V. Zharnitsky, Dynamical Systems Magazine, April, 2008)