Asymptotic behavior of the value functions of discrete-time discounted optimal control

We study deterministic discounted optimal control problems associated with discrete-time systems. It is shown that, for small discount rates, the cont rollability properties of the underlying system can guarantee the convergen ce of the discounted value function to the value function of the average yi eld. An application in the theory of exponential growth rates of discrete i nclusions is presented. This application motivates the analysis of infinite -horizon optimal control problems with running yields that are unbounded fr om below.