  In this paper, we revisit the sequential source coding framework to analyze
fundamental performance limitations of discrete-time stochastic control systems
subject to feedback data-rate constraints in finite-time horizon. The basis of
our results is a new characterization of the lower bound on the minimum
total-rate achieved by sequential codes subject to a total (across time)
distortion constraint and a computational algorithm that allocates optimally
the rate-distortion for any fixed finite-time horizon. This characterization
facilitates the derivation of analytical, non-asymptotic, and
finite-dimensional lower and upper bounds in two control-related scenarios. (a)
A parallel time-varying Gauss-Markov process with identically distributed
spatial components that is quantized and transmitted through a noiseless
channel to a minimum mean-squared error (MMSE) decoder. (b) A time-varying
quantized LQG closed-loop control system, with identically distributed spatial
components and with a random data-rate allocation. Our non-asymptotic lower
bound on the quantized LQG control problem, reveals the absolute minimum
data-rates for (mean square) stability of our time-varying plant for any fixed
finite time horizon. We supplement our framework with illustrative simulation
experiments.
