Stochastic stabilization of dynamical systems over communication channels
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We study stochastic stabilization of non-linear dynamical systems under information constraints. Towards this end, we develop a general method (based on dynamical systems theory, control theory, and probability) to derive fundamental lower bounds on the necessary channel capacity for stabilization over a discrete communication channel. The stability criterion considered is asymptotic mean stationarity (AMS). For noise-free channels, our method builds on the insight that the number of control inputs needed to accomplish a control task on a finite time interval is a measure for the necessary amount of information. A combination of this insight with certain characterizations of measure-theoretic entropy for dynamical systems yields a notion of entropy, which is tailored to derive lower bounds for asymptotic mean stationarity. For noisy channels, we develop a method to relate entropy and channel capacity through the strong converse to the channel coding theorem of information theory and optimal transport theory. Our fundamental bounds are consistent, and more refined when compared, with the bounds obtained earlier via methods based on differential entropy. 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.
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