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    Stationary Mean-Field Games with Congestion

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    Type
    Dissertation
    Authors
    Evangelista, David cc
    Advisors
    Gomes, Diogo A. cc
    Committee members
    Tempone, Raul cc
    Santamarina, Carlos cc
    Fusco, Nicola
    Program
    Applied Mathematics and Computational Science
    KAUST Department
    Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division
    Date
    2019-06-23
    Permanent link to this record
    http://hdl.handle.net/10754/655679
    
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    Abstract
    Mean-field games (MFG) are models of large populations of rational agents who seek to optimize an objective function that takes into account their state variables and the distribution of the state variable of the remaining agents. MFG with congestion model problems where the agents’ motion is hampered in high-density regions. First, we study radial solutions for first- and second-order stationary MFG with congestion on Rd. The radial case, which is one of the simplest non one-dimensional MFG, is relatively tractable. As we observe, the Fokker-Planck equation is integrable with respect to one of the unknowns. Consequently, we obtain a single equation substituting this solution into the Hamilton-Jacobi equation. For the first-order case, we derive explicit formulas; for the elliptic case, we study a variational formulation of the resulting equation. For the first case, we use our approach to compute numerical approximations to the solutions of the corresponding MFG systems. Next, we consider second-order stationary MFG with congestion and prove the existence of stationary solutions. Because moving in congested areas is difficult, agents prefer to move in non-congested areas. As a consequence, the model becomes singular near the zero density. The existence of stationary solutions was previously obtained for MFG with quadratic Hamiltonians thanks to a very particular identity. Here, we develop robust estimates that give the existence of a solution for general subquadratic Hamiltonians. Additionally, we study first-order stationary MFG with congestion with quadratic or power-like Hamiltonians. Using explicit examples, we illustrate two key difficulties: the lack of classical solutions and the existence of areas with vanishing densities. Our main contribution is a new variational formulation for MFG with congestion. With this formulation, we prove the existence and uniqueness of solutions. Finally, we devise a discretization that is combined with optimization algorithms to numerically solve various MFG with congestion.
    Citation
    Evangelista, D. (2019). Stationary Mean-Field Games with Congestion. KAUST Research Repository. https://doi.org/10.25781/KAUST-79GH9
    DOI
    10.25781/KAUST-79GH9
    ae974a485f413a2113503eed53cd6c53
    10.25781/KAUST-79GH9
    Scopus Count
    Collections
    Applied Mathematics and Computational Science Program; PhD Dissertations; Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division

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