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    On a Mean Field Optimal Control Problem

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    Type
    Preprint
    Authors
    Carrillo, Jose A.
    Pimentel, Edgard A.
    Voskanyan, Vardan K.
    Date
    2019-09-23
    Permanent link to this record
    http://hdl.handle.net/10754/660764
    
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    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.
    Sponsors
    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. 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.
    Publisher
    arXiv
    arXiv
    1909.10596
    Additional Links
    https://arxiv.org/pdf/1909.10596
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