Christoph Kawan: Stabilization of nonlinear deterministic systems over digital channels

Speaker: Christoph Kawan (Ludwig Maximilian University of Munich)

Title: Stabilization of Nonlinear Deterministic Systems over Digital Channels

Abstract: The data-rate theorem for linear systems has been extended in various directions by various groups of researchers. In their 2004 landmark paper, Nair et al. presented an extension to local stabilization of nonlinear deterministic systems around equilibrium points. In this talk, we take a broader perspective and replace the equilibrium by a compact set Lambda, invariant under a constant control input u_0. To analyze the associated local stabilization problem, we impose a hyperbolicity assumption on Lambda, which allows us to describe the dynamics close to Lambda under small time-dependent deviations from the constant input u_0, by using classical tools from hyperbolic dynamical systems. We use this description to derive a lower bound on the smallest average data rate needed for local stabilization to Lambda in terms of classical quantities from the ergodic theory of smooth dynamical systems. In two extremal cases, we are able to prove that our bound is sharp, i.e. equals the data rate limit. A general data-rate theorem, however, is still missing. As a matter of fact, any generalization of our results via similar methods leads to problems that are known to be difficult and are largely unsolved.