  In this paper, we consider solving discounted Markov Decision Processes
(MDPs) under the constraint that the resulting policy is stabilizing. In
practice MDPs are solved based on some form of policy approximation. We will
leverage recent results proposing to use Model Predictive Control (MPC) as a
structured policy in the context of Reinforcement Learning to make it possible
to introduce stability requirements directly inside the MPC-based policy. This
will restrict the solution of the MDP to stabilizing policies by construction.
The stability theory for MPC is most mature for the undiscounted MPC case.
Hence, we will first show in this paper that stable discounted MDPs can be
reformulated as undiscounted ones. This observation will entail that the
MPC-based policy with stability requirements will produce the optimal policy
for the discounted MDP if it is stable, and the best stabilizing policy
otherwise.
