Control systems of interacting objects modeled as a game against nature under a mean field approach

Carmen G. Higuera-Chan; Héctor Jasso-Fuentes; J. Adolfo Minjárez-Sosa

doi:10.3934/jdg.2017004

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2017, Volume 4, Issue 1: 59-74. Doi: 10.3934/jdg.2017004

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Control systems of interacting objects modeled as a game against nature under a mean field approach

1.
Departamento de Matemáticas, Universidad de Sonora, Rosales s/n, Col. Centro, 83000, Hermosillo, Sonora, México
2.
Departamento de Matemáticas, CINVESTAV-IPN, Apartado Postal 14-740, México D.F., 07000, México

* Corresponding author: aminjare@gauss.mat.uson.mx
* Corresponding author: aminjare@gauss.mat.uson.mx

Received: December 2015

Revised: December 2016

Published: March 2018

Work partially supported by Consejo Nacional de Ciencia y Tecnologia (CONACyT) under grants CB2015/254306 and CB2015/238045.

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Abstract

This paper deals with discrete-time stochastic systems composed of a large number of N interacting objects (a.k.a. agents or particles). There is a central controller whose decisions, at each stage, affect the system behavior. Each object evolves randomly among a finite set of classes, according to a transition law which depends on an unknown parameter. Such a parameter is possibly non observable and may change from stage to stage. Due to the lack of information and to the large number of agents, the control problem under study is rewritten as a game against nature according to the mean field theory; that is, we introduce a game model associated to the proportions of the objects in each class, whereas the values of the unknown parameter are now considered as "actions" selected by an opponent to the controller (the nature). Then, letting $N \to \infty $ (the mean field limit) and considering a discounted optimality criterion, the objective for the controller is to minimize the maximum cost, where the maximum is taken over all possible strategies of the nature.

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Mathematics Subject Classification: Primary: 93E20; Secondary: 60K35.

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