Stochastic stability and ergodicity of linear Gaussian systems controlled over discrete channels

We present necessary and sufficient conditions for stochastic stabilizability of unstable (non-stationary) noisy linear systems over channels with memory with feedback. Stochastic stability notions include recurrence, Birkhoff sample path ergodicity and asymptotic mean stationarity, and the existence of finite second moments. We extend recent results in the literature on noiseless and erasure channels for systems driven by possibly unbounded noise. Our constructive proof uses random-time state-dependent stochastic drift criteria for stabilization of Markov chains. For asymptotic mean stationarity, it is sufficient that the capacity of a channel is (strictly) greater than the sum of the logarithms of the unstable pole magnitudes for memoryless channels and a class of channels with memory. We also present the tightness of the sufficient condition under a technical condition. We provide sufficiency conditions for the existence of finite average second moments, for such systems driven by unbounded noise, which had not been studied in the literature to our knowledge. Comparison with relevant results in the literature is presented.