## Abstract

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→∞.

Original language | English |
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Journal | Annals of Operations Research |

DOIs | |

State | Accepted/In press - 2024 |

### Bibliographical note

Publisher Copyright:© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2024.

## Keywords

- Discounted criterion
- Mean field theory
- Semi-Markov control problems
- Systems of interacting objects
- Unbounded costs