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This paper discusses the functional stability of closed-loop Markov Chainsunder optimal policies resulting from a discounted optimality criterion,forming Markov Decision Processes (MDPs). We investigate the stability of MDPsin the sense of probability measures (densities) underlying the statedistributions and extend the dissipativity theory of Economic Model PredictiveControl in order to characterize the MDP stability. This theory requires aso-called storage function satisfying a dissipativity inequality. In theprobability measures space and for the discounted setting, we introduce newdissipativity conditions ensuring the MDP stability. We then use finite-horizonoptimal control problems in order to generate valid storage functionals. Inpractice, we propose to use Q-learning to compute the storage functionals.

Functional Stability of Discounted Markov Decision Processes Using Economic MPC Dissipativity Theory