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

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

doi:10.3934/jdg.2017004

Cited by 12 publications

(2 citation statements)

References 22 publications

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“…A consumption-investment example is discussed there. In [19] the same authors treat the unknown disturbance as a game against nature. [26] already start from a discrete-time mean-field control problem with finite time horizon.…”

Section: Introductionmentioning

confidence: 99%

Mean Field Markov Decision Processes

Bäuerle¹

2021

Preprint

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We consider mean-field control problems in discrete time with discounted reward, infinite time horizon and compact state and action space. The existence of optimal policies is shown and the limiting mean-field problem is derived when the number of individuals tends to infinity. Moreover, we consider the average reward problem and show that the optimal policy in this mean-field limit is ε-optimal for the discounted problem if the number of individuals is large and the discount factor close to one. This result is very helpful, because it turns out that in the special case when the reward does only depend on the distribution of the individuals, we obtain a very interesting subclass of problems where an average reward optimal policy can be obtained by first computing an optimal measure from a static optimization problem and then achieving it with Markov Chain Monte Carlo methods. We give two applications: Avoiding congestion an a graph and optimal positioning on a market place which we solve explicitly.

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Section: Introductionmentioning

confidence: 99%

Mean Field Markov Decision Processes

Bäuerle¹

2021

Preprint

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“…The past few years have witnessed many constructive and meaningful applications of mean‐field problems in various fields, such as finance, economics, games, and engineering, especially, in optimal control theory [5–10]. Continuous‐time MF‐LQ optimal control problems have been extensively studied with productive results [11–14].…”

Section: Introductionmentioning

confidence: 99%

Discrete‐time mean‐field stochastic linear‐quadratic optimal control problem with finite horizon

Song

Liu

2020

Asian Journal of Control

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In this paper we consider the discrete‐time mean‐field stochastic linear‐quadratic (MF‐LQ) optimal control problem with indefinite weighting matrices. First, we establish the maximum principle, and by the solvability of mean‐field forward‐backward stochastic difference equations derived from the maximum principle, we characterize the existence of the open‐loop optimal control for the MF‐LQ problem. Then, by virtue of introducing the linear matrix inequalities condition, we obtain the solvability of the generalized difference Riccati equations (GDREs). Moreover, we show that the indefinite MF‐LQ problem is well‐posed if and only if the GDREs are solvable. Finally, a numerical example is used to show the effectiveness of the obtained results.

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Approximation and Mean Field Control of Systems of Large Populations

Higuera-Chan

2021

Advances in Probability and Mathematical Statistics

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

Abstract: Systems of interacting objects, mean field theory, discounted criterion, games against nature, minimax control. Work partially supported by Consejo Nacional de Ciencia y Tecnologia (CONACyT) under grants CB2015/254306 and CB2015/238045.

Cited by 12 publications

References 22 publications

Mean Field Markov Decision Processes

Mean Field Markov Decision Processes

Discrete‐time mean‐field stochastic linear‐quadratic optimal control problem with finite horizon

Approximation and Mean Field Control of Systems of Large Populations

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