Introduction

Abstract

The dynamical theory of control attempts the integration of concepts and ideas from dynamical systems and control theory into a framework that sheds new light on both areas. In this context, the key control theoretic notions are control sets, chain control sets, linearization, and spectrum, while the basic concepts from the theory of dynamical systems are topological and chain recurrence, flows on vector bundles, and Lyapunov exponents. (One could add ergodic theory to this list, but the reader will notice that our integration of ergodic theory is quite preliminary, and where it plays a major role in our presentation, see Chapter 5, one could actually do without it; compare Grüne [143].) The link between control and dynamical systems theory is provided by the control flow, an infinite dimensional dynamical system that lives on the product space of admissible inputs and state space. The main topics in control theory that we study using the control-dynamic link are controllability regions and their domains of attraction, robust stability and stability radii, and open loop and feedback stabilization for nonlinear (and linear) systems. Applications to dynamical systems include (bounded) time varying perturbations of nominal systems, new spectral concepts (Morse spectrum), and persistence and continuity results for attractors and their spectra.KeywordsVector BundleLyapunov ExponentNominal SystemLocal AccessibilityStability RadiusThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.