Lyapunov Spectrum and Control Sets

Mathematical control theory analyzes systems with external, time dependent inputs. The present work concentrates on controllability questions for systems described by ordinary differential equations. This fundamental property of control systems is usually studied via differential geometric or functional-analytic methods.Here, a new approach is pioneered: controllability properties are derived from properties of Lyapunov exponents of the linearized system at a singular point. Using the theory of stable and unstable manifolds for nonautonomous differential equations combined with local accessibility properties, a number of results for the existence and location of completely controllable subsets, i.e., control sets, are given. This also opens a perspective for local bifurcation theory of control systems. Recently, this topic has found lively interest, motivated by technical applications and also by parallel developments in the theory of random differential systems.