Existence of weak solutions to stationary mean-field games through variational inequalities

SIAM Journal on Mathematical Analysis

Volumn 50, Issue 6, 2018, Pages 5969-6006

  Ferreira, Rita ^a   Gomes, Diogo ^a  

^a KING ABDULLAH UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Saudi Arabia)

Author keywords

Mean field games; Stationary problems; Variational inequalities

Indexed keywords

MATHEMATICAL MODELS; MATHEMATICAL TECHNIQUES;

EXISTENCE OF SOLUTIONS; EXISTENCE OF WEAK SOLUTIONS; MEAN FIELD GAMES; MONOTONICITY; NONLOCAL; STATIONARY PROBLEM; VARIATIONAL INEQUALITIES; WEAK SOLUTION;

VARIATIONAL TECHNIQUES;

EID: 85060554030     PISSN: 00361410     EISSN: 10957154     Source Type: Journal    
DOI: 10.1137/16M1106705     Document Type: Article

References (57)

1. Y. ACHDOU, Finite difference methods for mean field games, in Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications, Springer, Berlin, 2013, pp. 1-47.
2. N. ALMAYOUF et al., Existence of positive solutions for an approximation of stationary meanfield games, Involve, 10(2016), pp. 473-493.
3. N. ALMULLA, R. FERREIRA, AND D. GOMES, Two numerical approaches to stationary meanfield games, Dyn. Games Appl., 7(2017), pp. 657-682.
4. M. BARDI, Explicit solutions of some linear-quadratic mean field games, Netw. Heterog. Media, 7(2012), pp. 243-261.
5. M. BARDI AND F. PRIULI, LQG mean-field games with ergodic cost, in Proceedings of the 52nd IEEE Conference on Decision and Control, Florence, IEEE, Piscataway, NJ, 2013, pp. 2493-2498.
6. M. BARDI AND F. S. PRIULI, Linear-quadratic N-person and mean-field games with ergodic cost, SIAM J. Control Optim., 52(2014), pp. 3022-3052.
7. A. BENSOUSSAN, J. FREHSE, AND P. YAM, Mean Field Games and Mean Field Type Control Theory, SpringerBriefs Math., Springer, New York, 2013.
8. P. CARDALIAGUET, Notes on Mean-Field Games, manuscript.
9. P. CARDALIAGUET, Long time average of first order mean field games and weak KAM theory, Dyn. Games Appl., 3(2013), pp. 473-488.
10. P. CARDALIAGUET, Weak solutions for first order mean-field games with local coupling, in Analysis and Geometry in Control Theory and its Applications, Springer INdAM Ser. 11, Cham, Switzerland, 2015, pp. 111-158.
11. P. CARDALIAGUET, P. GARBER, A. PORRETTA, AND D. TONON, Second order mean field games with degenerate diffusion and local coupling, NoDEA Nonlinear Differential Equations Appl., 22(2015), pp. 1287-1317.
12. P. CARDALIAGUET AND P. J. GRABER, Mean field games systems of first order, ESAIM Control Optim. Calc. Var., 21(2015), pp. 690-722.
13. P. CARDALIAGUET, J.-M. LASRY, P.-L. LIONS, AND A. PORRETTA, Long time average of mean field games, Netw. Heterog. Media, 7(2012), pp. 279-301.
14. P. CARDALIAGUET, J.-M. LASRY, P.-L. LIONS, AND A. PORRETTA, Long time average of mean field games with a nonlocal coupling, SIAM J. Control Optim., 51(2013), pp. 3558-3591.
15. R. CARMONA, F. DELARUE, AND A. LACHAPELLE, Control of McKean-Vlasov dynamics versus mean field games, Math. Financ. Econ., 7(2013), pp. 131-166.
16. R. CARMONA AND D. LACKER, A probabilistic weak formulation of mean field games and applications, Ann. Appl. Probab., 25(2015), pp. 1189-1231.
17. J. DIEUDONNÉ, Foundations of Modern Analysis, Pure Appl. Math. 10 pt. 1, Academic Press, New York, 1969.
18. D. EVANGELISTA, R. FERREIRA, D. GOMES, L. NURBEKYAN, AND V. VOSKANYAN, First-order, stationary mean-field games with congestion, Nonlinear Anal., 173(2018), pp. 37-74.
19. D. EVANGELISTA, D. GOMES, AND L. NURBEKYAN, Radially symmetric mean-field games with congestion, in Proceedings of the 2017 IEEE 56th Annual Conference on Decision and Control (CDC), IEEE, Piscataway, NJ, 2017, pp. 3158-3163.
20. L. C. EVANS, Weak Convergence Methods for Nonlinear Partial Differential Equations, CBMS Reg. Conf. Ser. Math. 74, AMS, Providence, RI, 1990.
21. D. GOMES, Duality principles for fully nonlinear elliptic equations, in Trends in Partial Differential Equations of Mathematical Physics, Progr. Nonlinear Differential Equations Appl. 61, Birkhäuser, Basel, 2005, pp. 125-136.
22. D. GOMES, Generalized Mather problem and selection principles for viscosity solutions and Mather measures, Adv. Calc. Var., 1(2008), pp. 291-307.
23. D. GOMES AND H. MITAKE, Existence for stationary mean-field games with congestion and quadratic Hamiltonians, NoDEA Nonlinear Differential Equations Appl., 22(2015), pp. 1897-1910.
24. D. GOMES, L. NURBEKYAN, AND M. PRAZERES, Explicit solutions of one-dimensional, firstorder, stationary mean-field games with congestion, in Proceedings of the IEEE 55th Conference on Decision and Control, CDC 2016, IEEE, Piscataway, NJ, 2016, pp. 4534-4539.
25. D. GOMES, L. NURBEKYAN, AND M. PRAZERES, One-dimensional stationary mean-field games with local coupling, Dyn. Games Appl., 8(2017), pp. 315-351.
26. D. GOMES AND S. PATRIZI, Obstacle mean-field game problem, Interfaces Free Bound., 17(2015), pp. 55-68.
27. D. GOMES AND S. PATRIZI, Weakly coupled mean-field game systems, Nonlinear Anal., 144(2016), pp. 110-138.
28. D. GOMES, S. PATRIZI, AND V. VOSKANYAN, On the existence of classical solutions for stationary extended mean field games, Nonlinear Anal., 99(2014), pp. 49-79.
29. D. A. GOMES AND E. PIMENTEL, Time-dependent mean-field games with logarithmic nonlinearities, SIAM J. Math. Anal., 47(2015), pp. 3798-3812.
30. D. GOMES AND E. PIMENTEL, Local regularity for mean-field games in the whole space, Minimax Theory Appl., 1(2016), pp. 65-82.
31. D. GOMES, E. PIMENTEL, AND V. VOSKANYAN, Regularity Theory for Mean-Field Game Systems, SpringerBriefs Math., Springer, Cham, Switzerland, 2016.
32. D. GOMES AND E. A. PIMENTEL, Regularity for mean-field games systems with initial-initial boundary conditions: The subquadratic case, in Dynamics, Games and Science, CIM Ser. Math. Sci. 1, Springer, Cham, Switzerland, 2015, pp. 291-304.
33. D. GOMES, G. E. PIRES, AND H. SÁNCHEZ-MORGADO, A-priori estimates for stationary meanfield games, Netw. Heterog. Media, 7(2012), pp. 303-314.
34. D. GOMES AND R. RIBEIRO, Mean field games with logistic population dynamics, in Proceedings of the 52nd IEEE Conference on Decision and Control, Florence, December 2013, IEEE, Piscataway, NJ, 2013.
35. D. GOMES AND H. SÁNCHEZ MORGADO, A stochastic Evans-Aronsson problem, Trans. Amer. Math. Soc., 366(2014), pp. 903-929.
36. D. GOMES AND J. SAÚDE, Mean field games models-a brief survey, Dyn. Games Appl., 4(2014), pp. 110-154.
37. D. GOMES AND J. SAÚDE, Numerical methods for finite-state mean-field games satisfying a monotonicity condition, Appl. Math. Optim., to appear.
38. D. GOMES AND V. VOSKANYAN, Short-time existence of solutions for mean-field games with congestion, J. Lond. Math. Soc. (2), 92(2015), pp. 778-799.
39. D. A. GOMES AND V. K. VOSKANYAN, Extended deterministic mean-field games, SIAM J. Control Optim., 54(2016), pp. 1030-1055.
40. J. GRABER, Weak Solutions for Mean Field Games with Congestion, preprint, arXiv:1503.04733, 2015.
41. O. GUÉANT, A reference case for mean field games models, J. Math. Pures Appl. (9), 92(2009), pp. 276-294.
42. O. GUÉANT, Mean field games equations with quadratic Hamiltonian: A specific approach, Math. Models Methods Appl. Sci., 22(2012), 1250022.
43. M. HUANG, P. E. CAINES, AND R. P. MALHAMÉ, Large-population cost-coupled LQG problems with nonuniform agents: Individual-mass behavior and decentralized e-Nash equilibria, IEEE Trans. Automat. Control, 52(2007), pp. 1560-1571.
44. M. HUANG, R. P. MALHAMÉ, AND P. E. CAINES, Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle, Commun. Inf. Syst., 6(2006), pp. 221-251.
45. H. ISHII, H. MITAKE, AND H. V. TRAN, The vanishing discount problem and viscosity Mather measures. Part 1: The problem on a torus, J. Math. Pures Appl. (9), 108(2017), pp. 125-149.
46. H. ISHII, H. MITAKE, AND H. V. TRAN, The vanishing discount problem and viscosity Mather measures. Part 2: Boundary value problems, J. Math. Pures Appl. (9), 108(2017), pp. 261-305.
47. D. KINDERLEHRER AND G. STAMPACCHIA, An Introduction to Variational Inequalities and their Applications, Classics Appl. Math. 31, SIAM, Philadelphia, 2000.
48. D. LACKER, A general characterization of the mean field limit for stochastic differential games, Probab. Theory Related Fields, 165(2016), pp. 581-648.
49. J.-M. LASRY AND P.-L. LIONS, Jeux a champ moyen. I. Le cas stationnaire, C. R. Math. Acad. Sci. Paris, 343(2006), pp. 619-625.
50. J.-M. LASRY AND P.-L. LIONS, Jeux a champ moyen. II. Horizon fini et controle optimal, C. R. Math. Acad. Sci. Paris, 343(2006), pp. 679-684.
51. J.-M. LASRY AND P.-L. LIONS, Mean field games, Jpn. J. Math., 2(2017), pp. 229-260.
52. P. L. LIONS, Collége de France Course on Mean-Field Games, video.
53. L. NURBEKYAN, One-dimensional, non-local, first-order, stationary mean-field games with congestion: A Fourier approach, Discrete Contin. Dyn. Syst. Ser. B, 11(2018), pp. 963-990.
54. E. PIMENTEL AND V. VOSKANYAN, Regularity for second-order stationary mean-field games, Indiana Univ. Math. J., 66(2017), pp. 1-22.
55. A. PORRETTA, On the planning problem for the mean field games system, Dyn. Games Appl., 4(2014), pp. 231-256.
56. A. PORRETTA, Weak solutions to Fokker-Planck equations and mean field games, Arch. Ration. Mech. Anal., 216(2015), pp. 1-62.
57. V. K. VOSKANYAN, Some estimates for stationary extended mean field games, Dokl. Nats. Akad. Nauk Armen., 113(2013), pp. 30-36.