First-order, stationary mean-field games with congestion

Evangelista, David; Ferreira, Rita; Gomes, Diogo A.; Nurbekyan, Levon; Voskanyan, Vardan

doi:10.48550/arxiv.1710.01566

Cited by 2 publications

(10 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For first order MFG systems with congestion, weak solutions were shown to exist by A. Porretta and Y. Achdou [2], but classical solutions had not been obtained so far. Second order MFG systems with congestion were also studied in [1,8,9,10], where weak solutions and short-time existence result in the smooth setting have been obtained. Finally, for second order mean field type control problems with congestion, weak solutions were obtained in [3].…”

Section: Introductionmentioning

confidence: 99%

Classical and weak solutions to local first-order mean field games through elliptic regularity

Muñoz

2022

Ann. Inst. H. Poincaré C Anal. Non Linéaire

View full text Add to dashboard Cite

We study the regularity and well-posedness of the local, first-order forward-backward mean field games system, assuming a polynomially growing cost function and a Hamiltonian of quadratic growth. We consider systems and terminal data that are strictly monotone in density and study two different regimes depending on whether there exists a lower bound for the running cost function. The work relies on a transformation due to P.-L. Lions, which gives rise to an elliptic partial differential equation with oblique boundary conditions, that is strictly elliptic when the coupling is unbounded from below. In this case, we prove that the solution is smooth. When the problem is degenerate elliptic, we obtain existence and uniqueness of weak solutions analogous to those obtained by P. Cardaliaguet and P. J. Graber for the case of a terminal condition that is independent of the density. The weak solutions are shown to arise as viscous limits of classical solutions to strictly elliptic problems.

show abstract

Section: Introductionmentioning

confidence: 99%

Classical and weak solutions to local first-order mean field games through elliptic regularity

Muñoz

2022

Ann. Inst. H. Poincaré C Anal. Non Linéaire

View full text Add to dashboard Cite

show abstract

“…However, for a specific H it may still be possible to find a non-standard variational formulation for (1.1), (1.2). In [29], for instance, authors found a new variational formulation for (1.2) when ε = 0 and…”

Section: Introductionmentioning

confidence: 99%

“…, and 1 < α γ. The formulation in [29] yielded well-defined variational solutions that are unique and can be numerically calculated by optimization techniques.…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, in analogy with [9], we find a convex optimization formulation of this saddle point problem that is also new: Remark 4.7. Moreover, as we observe in Section 5, this convex optimization formulation is a generalization of the transformation in [29] that was used to solve (1.2) for H of the form (1.5) and parameter range 0 < α < 1 γ in the two-dimensional case.…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, when 1 < α γ the potential of this game is concave, and therefore Nash equilibria are the maximizers of this potential. Thus, in Section 5, we recover the variational principle in [29].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The variational structure and time-periodic solutions for mean-field games systems

Cirant,

Nurbekyan

2018

Preprint

Self Cite

View full text Add to dashboard Cite

Here, we observe that mean-field game (MFG) systems admit a twoplayer infinite-dimensional general-sum differential game formulation. We show that particular regimes of this game reduce to previously known variational principles. Furthermore, based on the game-perspective we derive new variational formulations for first-order MFG systems with congestion. Finally, we use these findings to prove the existence of time-periodic solutions for viscous MFG systems with a coupling that is not a non-decreasing function of density.

show abstract

First-order, stationary mean-field games with congestion

Cited by 2 publications

References 0 publications

Classical and weak solutions to local first-order mean field games through elliptic regularity

Classical and weak solutions to local first-order mean field games through elliptic regularity

The variational structure and time-periodic solutions for mean-field games systems

Contact Info

Product

Resources

About