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

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发表日期：

2024年4月8日

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M. Elena Martínez-Manzanares,
J. Adolfo Minjárez-Sosa

摘要

This paper is about optimal control problems associated to stochastic systems composed of a large number of N ( $N\sim \infty $ $N \sim \infty$ ) 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 $\mathcal{S}\mathcal{M}_{N}$ $S M_{N}$ 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\rightarrow \infty $ $N \to \infty$ (the mean field limit). Then we obtain the mean field control model $\mathcal{S}\mathcal{M}$ $S M$ , independent on N, which is easier to study than $\mathcal{S}\mathcal{M}_{N}.$ $S M_{N} .$ Our main objective is to show that an optimal policy $\pi _{*},$ $π_{*},$ under a discounted criterion, in $\mathcal{S}\mathcal{M}$ $S M$ has a good behavior in $\mathcal{S}\mathcal{M}_{N}.$ $S M_{N} .$ Specifically, we prove that $\pi _{*}$ $π_{*}$ is nearly discounted optimal in $\mathcal{S}\mathcal{M}_{N}$ $S M_{N}$ asymptotically as $N\rightarrow \infty .$ $N \to \infty .$