Lyapunov exponents of control flows

Abstract

The use of Lyapunov exponents in the theory of dynamical systems or stochastic systems is often based on Oseledeč's Multiplicative Ergodic Theorem. For control systems this is not possible, because each (sufficiently rich) control system contains dynamics that are not Lyapunov regular. In this paper we present an approach to study the Lyapunov spectrum of a nonlinear control system via ergodic theory of the associated control flow and its linearization. In particular, it turns out that all Lyapunov exponents are attained over so called chain control sets, and they are integrals of Lyapunov exponents on control sets with respect to flow invariant measures, whose support is contained in the lifts of control sets to U×M, where U is the space of admissible control functions and M is the state space of the system. For the linearization of control systems about rest points the extremal Lyapunov exponents are analyzed, which leads to precise criteria for the stabilization and destabilization of bilinear control systems, and to robustness results for linear systems. The last section is devoted to a nonlinear example, where we combine the analysis of the global controllability structure with local linearization results and Lyapunov exponents to obtain a complete picture of control, stabilization and robustness of the system.