Adaptive control of specially structured Markov chains

We consider Markov decision processes where the state at time n+1 is a function of the state at time n, the action at time n and the outcome of a random variable Y_{n+1}. The random variables Y_1, Y_2, Y_3, ... are independent and identically distributed with an incompletely known distribution. The class of problems considered includes the linear system with quadratic cost and a simple inventory control model. The minimal Bayesian expected total cost is determined or approximated. The strategy that takes, at each time, the action that is optimal if the estimated distribution is the true distribution, is studied.