Jointly Optimal LQG quantization and control policies for multi-dimensional linear Gaussian sources

For controlled Rⁿ-valued linear systems driven by Gaussian noise under quadratic cost criteria, we investigate the existence and the structure of optimal quantization and control policies. For a fully observed system, we show that an optimal quantization policy exists, provided that the quantizers allowed are ones which have convex codecells. Furthermore, optimal control policies are linear in the conditional estimate of the state. A form of separation and estimation applies. As a minor side result, towards obtaining the main results of the paper, structural results in the literature for optimal causal (zero-delay) quantization of Markov sources is extended to systems driven by control. For the partially observed case, structure of optimal coding and control policies is presented.