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Given a stochastic nonlinear system controlled over a possibly noisy communication channel, the article studies the largest class of channels for which there exist coding and control policies so that the closed-loop system is stochastically stable. The stability criterion considered is asymptotic mean stationarity (AMS). In this article, we develop a general method based on ergodic theory and probability to derive fundamental bounds on information transmission requirements leading to stabilization. Through this method, we develop a new notion of entropy which is tailored to derive lower bounds for AMS for both noise-free and noisy channels. The bounds obtained through probabilistic and ergodic-theoretic analysis are more refined in comparison with the bounds obtained earlier via information-theoretic methods. Moreover, our approach is more versatile in view of the models considered and allows for finer lower bounds when the AMS measure is known to admit further properties, such as moment bounds.

Invariance Properties of Controlled Stochastic Nonlinear Systems Under Information Constraints