Research Aticles

  1. On a mean-field optimal control problem, Nonlinear Analysis, Volume 199, October 2020, 112039.
    (joint work with Jose Carrillo and Edgard Pimentel)

    In this paper we consider a mean field optimal control problem with an aggregationdiffusion 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.

  2. First-order, stationary mean-field games with congestion (with ), Nonlinear Analysis, 173 (2018), 37-74
    (joint work with Diogo Gomes, Rita Ferreira, Levon Nurbekyan, and David Evangelista)

    Mean-field games (MFGs) are models for large populations of competing rational agents that seek to optimize a suitable functional. In the case of congestion, this functional takes into account the difficulty of moving in high-density areas. Here, we study stationary MFGs with congestion with quadratic or power-like Hamiltonians. First, using explicit examples, we illustrate two main difficulties: the lack of classical solutions and the existence of areas with vanishing densities. Our main contribution is a new variational formulation for MFGs with congestion. With this formulation, we prove the existence and uniqueness of solutions. Finally, we consider applications to numerical methods.

  3. Regularity for second-order mean-field games Indiana University Mathematics Journal, Vol. 66, No. 1 (2017) , pp. 1-22.
    (joint work with Edgard Pimentel)

    In this paper, we prove the existence of classical solutions for second-order stationary mean-field game systems. These arise in ergodic (mean-field) optimal control, convex degenerate problems in calculus of variations, and in the study of long-time behavior of time-dependent mean-field games. Our argument is based on the interplay between the regularity of solutions of the Hamilton-Jacobi equation in terms of the solutions of the Fokker-Planck equation and vice-versa. Because we consider different classes of couplings, distinct techniques are used to obtain a priori estimates for the density. In the case of polynomial couplings, we resort to an iterative method. An integral method builds upon the properties of the logarithmic function in the setting of logarithmic nonlinearities. This work extends substantially previous results by allowing for more general classes of Hamiltonians and mean-field assumptions.

  4. Extended deterministic mean-field games, SIAM Journal on Control and Optimization, 54 (2016), 2/4 no. 2, 1030-1055.
    (joint work with Diogo Gomes)

    In this paper, we consider mean-field games where the interaction of each player with the mean field takes into account not only the states of the players but also their collective behavior. To do so, we develop a random variable framework that is particularly convenient for these problems. We prove an existence result for extended mean-field games and establish uniqueness conditions. In the last section, we consider the Master Equation and discuss properties of its solutions.

  5. Short-time existence of solutions for mean-field games with congestion, Journal of the London Mathematical Society, (2) 92 (2015), no. 3, 778-799.
    (joint work with Diogo Gomes)

    We consider time-dependent mean-field games with congestion that are given by a Hamilton–Jacobi equation coupled with a Fokker–Planck equation. These models are motivated by crowd dynamics in which agents have difficulty moving in high-density areas. The congestion effects make the Hamilton–Jacobi equation singular. The uniqueness of solutions for this problem is well understood; however, the existence of classical solutions was only known in very special cases, stationary problems with quadratic Hamiltonians and some time-dependent explicit examples. Here, we demonstrate the short-time existence of smooth solutions for sub-quadratic Hamiltonians.

  6. On the existence of classical solutions for stationary extended mean-field games, Nonlinear Analysis, Theory, Methods & Applications 99 (2014), 49-79.
    (joint work with Diogo Gomes and Stefania Patrizi)

    In this paper we consider extended stationary mean-field games, that is mean-field games which depend on the velocity field of the players. We prove various a-priori estimates which generalize the results for quasi-variational mean-field games in Gomes et al. (2012). In addition we use adjoint method techniques to obtain higher regularity bounds. Then we establish the existence of smooth solutions under fairly general conditions by applying the continuity method. When applied to standard stationary mean-field games as in Lasry and Lions (2006), Gomes and Sanchez-Morgado (2011) or Gomes et al. (2012) this paper yields various new estimates and regularity properties not available previously. We discuss additionally several examples where the existence of classical solutions can be proved.

  7. Some estimates for stationary extended mean field games, Proceedings of the YSU, 2013, Issue 1, 24–31.
    (joint work with Diogo Gomes)

    Mean field games theory is a recent area of study introduced by Lions and Lasry in a series of seminal papers in 2006. Mean field games model situations of competition between large number of rational agents that play noncooperative dynamic games under certain symmetry assumptions. In this paper we consider quasivariational mean field games system with additional dependence on a velocity field of the players. We obtain certain estimates for the solutions to this system. In a forthcoming paper we intend to obtain an existence result using this kind of estimates and continuation method.

Books

  1. Regularity Theory for Mean-Field Game Systems, Springer Briefs in Mathematics, 2016.
    (joint work with Diogo Gomes and Edgard Pimentel)

    This book brings together several recent developments on the regularity theory for mean-field game systems. We detail several classes of methods and present a concise overview of the main techniques developed in the last few years. Most of the forthcoming material deals with simple and computation-friendly examples; this is intended to unveil the main ideas behind the methods rather than focus on the technicalities of particular cases. The choice of topics presented here reflects the authors’ perspective on this fastgrowing field of research; it is by no means exhaustive or intended as a complete account of the theory. Rather—and in the best scenario—it serves as an introduction to the material available in scientific papers.