On a mean field optimal control problem

José A. Carrillo; Edgard A. Pimentel; Vardan K. Voskanyan

doi:10.1016/j.na.2020.112039

On a mean field optimal control problem

José A. Carrillo, Edgard A. Pimentel, Vardan K. Voskanyan

Research output: Contribution to journal › Article › peer-review

7 Scopus citations

Abstract

In this paper we consider a mean field optimal control problem with an aggregation–diffusion constraint, where agents interact through a potential, in the presence of a Gaussian noise term. Our analysis focuses on a PDE system coupling a Hamilton–Jacobi and a Fokker–Planck equation, describing the optimal control aspect of the problem and the evolution of the population of agents, respectively. The main contribution of the paper is a result on the existence of solutions for the aforementioned system. We notice this model is in close connection with the theory of mean-field games systems. However, a distinctive feature concerns the nonlocal character of the interaction; it affects the drift term in the Fokker–Planck equation as well as the Hamiltonian of the system, leading to new difficulties to be addressed.

Original language	English (US)
Pages (from-to)	112039
Journal	Nonlinear Analysis, Theory, Methods and Applications
Volume	199
DOIs	https://doi.org/10.1016/j.na.2020.112039
State	Published - Jun 30 2020
Externally published	Yes

Bibliographical note

KAUST Repository Item: Exported on 2022-06-14
Acknowledgements: JAC was partially supported by the EPSRC grant number EP/P031587/1. EAP was partially supported by FAPERJ (# E26/200.002/2018), CNPq-Brazil (#433623/2018-7 and #307500/2017-9) and Instituto Serrapilheira (#1811-25904). VKV was partially supported by FCT – Fundação para a Ciência e a Tecnologia, I.P. through projects PTDC/MAT-PUR/28686/2017 and by CMUC – UID/MAT/00324/2013, funded by the Portuguese government through FCT and co-funded by the European Regional Development Fund through Partnership Agreement PT2020. We would like to acknowledge the Institute Mittag-Leffler, Imperial College London and King Abdullah University of Science and Technology for hosting us and providing with constant help and vivid research environment.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1016/j.na.2020.112039

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title = "On a mean field optimal control problem",

abstract = "In this paper we consider a mean field optimal control problem with an aggregation–diffusion constraint, where agents interact through a potential, in the presence of a Gaussian noise term. Our analysis focuses on a PDE system coupling a Hamilton–Jacobi and a Fokker–Planck equation, describing the optimal control aspect of the problem and the evolution of the population of agents, respectively. The main contribution of the paper is a result on the existence of solutions for the aforementioned system. We notice this model is in close connection with the theory of mean-field games systems. However, a distinctive feature concerns the nonlocal character of the interaction; it affects the drift term in the Fokker–Planck equation as well as the Hamiltonian of the system, leading to new difficulties to be addressed.",

author = "Carrillo, {Jos{\'e} A.} and Pimentel, {Edgard A.} and Voskanyan, {Vardan K.}",

note = "KAUST Repository Item: Exported on 2022-06-14 Acknowledgements: JAC was partially supported by the EPSRC grant number EP/P031587/1. EAP was partially supported by FAPERJ (# E26/200.002/2018), CNPq-Brazil (#433623/2018-7 and #307500/2017-9) and Instituto Serrapilheira (#1811-25904). VKV was partially supported by FCT – Funda{\c c}{\~a}o para a Ci{\^e}ncia e a Tecnologia, I.P. through projects PTDC/MAT-PUR/28686/2017 and by CMUC – UID/MAT/00324/2013, funded by the Portuguese government through FCT and co-funded by the European Regional Development Fund through Partnership Agreement PT2020. We would like to acknowledge the Institute Mittag-Leffler, Imperial College London and King Abdullah University of Science and Technology for hosting us and providing with constant help and vivid research environment. This publication acknowledges KAUST support, but has no KAUST affiliated authors.",

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AU - Pimentel, Edgard A.

AU - Voskanyan, Vardan K.

N1 - KAUST Repository Item: Exported on 2022-06-14 Acknowledgements: JAC was partially supported by the EPSRC grant number EP/P031587/1. EAP was partially supported by FAPERJ (# E26/200.002/2018), CNPq-Brazil (#433623/2018-7 and #307500/2017-9) and Instituto Serrapilheira (#1811-25904). VKV was partially supported by FCT – Fundação para a Ciência e a Tecnologia, I.P. through projects PTDC/MAT-PUR/28686/2017 and by CMUC – UID/MAT/00324/2013, funded by the Portuguese government through FCT and co-funded by the European Regional Development Fund through Partnership Agreement PT2020. We would like to acknowledge the Institute Mittag-Leffler, Imperial College London and King Abdullah University of Science and Technology for hosting us and providing with constant help and vivid research environment. This publication acknowledges KAUST support, but has no KAUST affiliated authors.

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N2 - In this paper we consider a mean field optimal control problem with an aggregation–diffusion constraint, where agents interact through a potential, in the presence of a Gaussian noise term. Our analysis focuses on a PDE system coupling a Hamilton–Jacobi and a Fokker–Planck equation, describing the optimal control aspect of the problem and the evolution of the population of agents, respectively. The main contribution of the paper is a result on the existence of solutions for the aforementioned system. We notice this model is in close connection with the theory of mean-field games systems. However, a distinctive feature concerns the nonlocal character of the interaction; it affects the drift term in the Fokker–Planck equation as well as the Hamiltonian of the system, leading to new difficulties to be addressed.

AB - In this paper we consider a mean field optimal control problem with an aggregation–diffusion constraint, where agents interact through a potential, in the presence of a Gaussian noise term. Our analysis focuses on a PDE system coupling a Hamilton–Jacobi and a Fokker–Planck equation, describing the optimal control aspect of the problem and the evolution of the population of agents, respectively. The main contribution of the paper is a result on the existence of solutions for the aforementioned system. We notice this model is in close connection with the theory of mean-field games systems. However, a distinctive feature concerns the nonlocal character of the interaction; it affects the drift term in the Fokker–Planck equation as well as the Hamiltonian of the system, leading to new difficulties to be addressed.

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On a mean field optimal control problem

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