Stationary mean-field games with logistic effects
Tóm tắt
In its standard form, a mean-field game is a system of a Hamilton-Jacobi equation coupled with a Fokker-Planck equation. In the context of population dynamics, it is natural to add to the Fokker-Planck equation features such as seeding, birth, and non-linear death rates. Here, we consider a logistic model for the birth and death of the agents. Our model applies to situations in which crowding increases the death rate. The new terms in this model require novel ideas to obtain the existence of a solution. Here, the main difficulty is the absence of monotonicity. Therefore, we construct a regularized model, establish a priori estimates for the solution, and then use a limiting argument to obtain the result.
Tài liệu tham khảo
Almulla, N., Ferreira, R., Gomes, D.: Two numerical approaches to stationary mean-field games. Dyn. Games Appl. 7(4), 657–682 (2017)
Bensoussan, A., Frehse, J., Yam, P.: Mean field games and mean field type control theory. Springer briefs in mathematics. Springer, New York (2013)
Boccardo, L., Orsina, L., Porretta, A.: Strongly coupled elliptic equations related to mean-field games systems. J. Differ. Equ. 261(3), 1796–1834 (2016)
Briceño Arias, L.M., Kalise, D., Silva, F.J.: Proximal methods for stationary mean field games with local couplings. SIAM J. Control Optim. 56(2), 801–836 (2018)
Carmona, R., Delarue, F.: Probabilistic theory of mean field games with applications I–II. Springer, Berlin (2018)
Evangelista, D., Ferreira, R., Gomes, D., Nurbekyan, L., Voskanyan, V.: First-order, stationary mean-field games with congestion. Nonlinear Anal. 173, 37–74 (2018)
Evangelista, D., Gomes, D.: On the existence of solutions for stationary mean-field games with congestion. J. Dynamics Differ. Equ. 30(4), 1365–1388 (2018)
Evans, L.C.: Adjoint and compensated compactness methods for Hamilton-Jacobi PDE. Arch. Ration. Mech. Anal. 197(3), 1053–1088 (2010)
Ferreira, R., Gomes, D.: Existence of weak solutions to stationary mean-field games through variational inequalities. SIAM J. Math. Anal. 50(6), 5969–6006 (2018)
Ferreira, R., Gomes, D., Tada, T.: Existence of weak solutions to first-order stationary mean-field games with Dirichlet conditions. Proc. Am. Math. Soc. 147(11), 4713–4731 (2019)
Gomes, D., Mitake, H.: Existence for stationary mean-field games with congestion and quadratic Hamiltonians. NoDEA Nonlinear Differ. Equ. Appl. 22(6), 1897–1910 (2015)
Gomes, D., Nurbekyan, L., Pimentel, E.: Economic models and mean-field games theory. Publicações Matemáticas do IMPA. [IMPA Mathematical Publications]. Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2015. 30\({^{{}}{\rm {o}}}\) Colóquio Brasileiro de Matemática. [30th Brazilian Mathematics Colloquium]
Gomes, D., Nurbekyan, L., Prazeres, M.: One-dimensional stationary mean-field games with local coupling. Dyn, Games and Applications (2017)
Gomes, D., Patrizi, S.: Obstacle mean-field game problem. Interfaces Free Bound. 17(1), 55–68 (2015)
Gomes, D., Patrizi, S., Voskanyan, V.: On the existence of classical solutions for stationary extended mean field games. Nonlinear Anal. 99, 49–79 (2014)
Gomes, D., Pimentel, E., Sánchez-Morgado, H.: Time-dependent mean-field games in the subquadratic case. Comm. Part. Differ. Equ. 40(1), 40–76 (2015)
Gomes, D., Pimentel, E., Sánchez-Morgado, H.: Time-dependent mean-field games in the superquadratic case. ESAIM Control Optim. Calc. Var. 22(2), 562–580 (2016)
Gomes, D., Pimentel, E., Voskanyan, V.: Regularity theory for mean-field game systems. Springer Briefs in Mathematics. Springer, Cham (2016)
Gomes, D., Pires, G.E., Sánchez-Morgado, H.: A-priori estimates for stationary mean-field games. Netw. Heterog. Media 7(2), 303–314 (2012)
Gomes, D., Ribeiro, R.: Mean field games with logistic population dynamics. 52nd IEEE Conference on Decision and Control (Florence, December 2013) (2013)
Gomes, D., Sánchez Morgado, H.: A stochastic Evans-Aronsson problem. Trans. Am. Math. Soc. 366(2), 903–929 (2014)
Huang, M., Caines, P.E., Malhamé, R.P.: Large-population cost-coupled LQG problems with nonuniform agents: individual-mass behavior and decentralized \(\epsilon \)-Nash equilibria. IEEE Trans. Automat. Control 52(9), 1560–1571 (2007)
Huang, M., Malhamé, R.P., Caines, P.E.: Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. Commun. Inf. Syst. 6(3), 221–251 (2006)
Lasry, J.-M., Lions, P.-L.: Jeux à champ moyen. I. Le cas stationnaire. CR Math. Acad. Sci. Paris 343(9), 619–625 (2006)
Lasry, J.-M., Lions, P.-L.: Jeux à champ moyen. II. Horizon fini et contrôle optimal. CR Math. Acad. Sci. Paris 343(10), 679–684 (2006)
Lions, P.L.: Collège de France course on mean-field games. 2007–2011
Mészáros, A.R., Silva, F.J.: On the variational formulation of some stationary second-order mean field games systems. SIAM J. Math. Anal. 50(1), 1255–1277 (2018)
Nurbekyan, L.: One-dimensional, non-local, first-order stationary mean-field games with congestion: a Fourier approach. Discrete Contin. Dyn. Syst. Ser. S 11(5), 963–990 (2018)
Pimentel, E., Voskanyan, V.: Regularity for second-order stationary mean-field games. Indiana Univ. Math. J. 66(1), 1–22 (2017)