TY - JOUR

T1 - Semi-Markov control models for systems of large populations of interacting objects with possible unbounded costs

T2 - a mean field approach

AU - Martínez-Manzanares, M. Elena

AU - Minjárez-Sosa, J. Adolfo

N1 - Publisher Copyright:
© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2024.

PY - 2024

Y1 - 2024

N2 - 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 SMN 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 SMN. Our main objective is to show that an optimal policy π∗, under a discounted criterion, in SM has a good behavior in SMN. Specifically, we prove that π∗ is nearly discounted optimal in SMN asymptotically as N→∞.

AB - 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 SMN 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 SMN. Our main objective is to show that an optimal policy π∗, under a discounted criterion, in SM has a good behavior in SMN. Specifically, we prove that π∗ is nearly discounted optimal in SMN asymptotically as N→∞.

KW - Discounted criterion

KW - Mean field theory

KW - Semi-Markov control problems

KW - Systems of interacting objects

KW - Unbounded costs

UR - http://www.scopus.com/inward/record.url?scp=85189491703&partnerID=8YFLogxK

U2 - 10.1007/s10479-024-05937-2

DO - 10.1007/s10479-024-05937-2

M3 - Artículo

AN - SCOPUS:85189491703

SN - 0254-5330

JO - Annals of Operations Research

JF - Annals of Operations Research

ER -