Ergodicity of Controlled Stochastic Nonlinear Systems under Information Constraints: Refined Bounds via Splitting

This paper considers the problem of stabilizing a discrete-time non-linear stochastic system over a finite capacity noiseless channel. In the literature, it has been established that under technical assumptions, the channel capacity must not be smaller than the logarithm of the determinant of the system linearization, averaged over noise and ergodic state measures. However, this result is not tight for many systems as stable directions may hide the volume growth arising from unstable directions. In this paper, utilizing a stochastic and geometric analysis, we introduce a method which allows us to consider stochastic growth among subsets of coordinates. Accordingly, under the asymptotic ergodicity stability criterion, we establish a lower bound on information transmission requirements for the existence of a coding and control policy which renders the closed-loop system stochastically stable. We establish that for systems with a stochastically stable component, it suffices to consider only the unstable dimensions, providing a refinement on the general channel capacity bound for a large class of systems.