Semi-Markov control models for systems of large populations of interacting objects with possible unbounded costs: a mean field approach

This paper is about optimal control problems associated to stochastic systems composed of a large number of N (N∼∞) interacting objects (e.g., particles, agents, data, etc.) evolving among a finite or countable set of classes or categories according to a semi-Markov process. Such systems are modeled by a control model SM_N where the states are vectors whose components are the proportions of objects in each class. Since N is too large, from a practical point of view, it is almost impossible to obtain a solution of the control problem. Under this setting, we apply a mean field approach which consists of letting N→∞ (the mean field limit). Then we obtain the mean field control model SM, independent on N, which is easier to study than SM_N. Our main objective is to show that an optimal policy π_∗, under a discounted criterion, in SM has a good behavior in SM_N. Specifically, we prove that π_∗ is nearly discounted optimal in SM_N asymptotically as N→∞.