Approximation of Stationary Control Policies by Quantized Control in Markov Decision Processes

We consider the problem of approximating optimal stationary control policies by quantized control. Stationary quantizer policies are introduced and it is shown that such policies are epsilon-optimal among stationary policies under mild technical conditions. Quantitative bounds on the approximation error in terms of the rate of the approximating quantizers are also derived. Thus, one can search for epsilon-optimal policies within quantized control policies. These pave the way for applications in optimal design of networked control systems where controller actions need to be quantized, as well as for a new computational method for the generation of approximately optimal Markov decision policies in general (Borel) state and action spaces for both discounted cost and average cost infinite horizon optimal control problems.