Invariance and stationarity in non-linear networked control: Deterministic and stochastic formulations

In this paper we compare two recent threads of research on the stabilization of non-linear control systems under information constraints. In a deterministic setting, characterizations of the smallest bit rate of a noiseless channel connecting a state coder to a controller have been obtained, above which local stabilization or set-invariance is possible. These characterizations use quantities similar to topological entropy in dynamical systems and lead to estimates of the corresponding bit rates in terms of Lyapunov exponents. In a stochastic setting, a decisive quantity is the channel capacity. For different notions of stochastic stability, namely stationarity, ergodicity and asymptotic mean stationarity, estimates for the smallest capacity, above which stabilization in one of these senses can be accomplished, are available. We review the results of the deterministic and the stochastic theory, highlight their analogies and compare the corresponding methods of proof.