no. 11-12
Partial Differential Equations
On the planning problem for a class of Mean Field Games
[Sur le problème de planification pour une classe de jeux à champ moyen]
Présenté par : Pierre-Louis Lions
Porretta, Alessio1
1 Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Roma, Italy
Comptes Rendus. Mathématique, Tome 351 (2013) no. 11-12, pp. 457-462

We give a result of existence and uniqueness of weak solutions to the planning problem for a class of Mean Field Games. This is a kind of optimal transportation problem consisting in the exact controllability at time T of Fokker–Planck equations obtained using drifts arising as the optimal feedbacks from a coupled backward Hamilton–Jacobi–Bellman equation.

Nous donnons un résultat dʼexistence et dʼunicité des solutions faibles du problème de planification pour une classe de jeux à champ moyen. Il sʼagit dʼun problème de transport optimal qui consiste en la contrôlabilité exacte au temps T de lʼéquation de Fokker–Planck en utilisant des champs obtenus comme loi feedback optimale dʼune équation de Hamilton–Jacobi–Bellman couplée.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2013.07.004

Porretta, Alessio 1

1 Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Roma, Italy 
@article{CRMATH_2013__351_11-12_457_0,
     author = {Porretta, Alessio},
     title = {On the planning problem for a class of {Mean} {Field} {Games}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {457--462},
     year = {2013},
     publisher = {Elsevier},
     volume = {351},
     number = {11-12},
     doi = {10.1016/j.crma.2013.07.004},
     language = {en},
     url = {https://www.numdam.org/articles/10.1016/j.crma.2013.07.004/}
}
TY  - JOUR
AU  - Porretta, Alessio
TI  - On the planning problem for a class of Mean Field Games
JO  - Comptes Rendus. Mathématique
PY  - 2013
SP  - 457
EP  - 462
VL  - 351
IS  - 11-12
PB  - Elsevier
UR  - https://www.numdam.org/articles/10.1016/j.crma.2013.07.004/
DO  - 10.1016/j.crma.2013.07.004
LA  - en
ID  - CRMATH_2013__351_11-12_457_0
ER  - 
%0 Journal Article
%A Porretta, Alessio
%T On the planning problem for a class of Mean Field Games
%J Comptes Rendus. Mathématique
%D 2013
%P 457-462
%V 351
%N 11-12
%I Elsevier
%U https://www.numdam.org/articles/10.1016/j.crma.2013.07.004/
%R 10.1016/j.crma.2013.07.004
%G en
%F CRMATH_2013__351_11-12_457_0
Porretta, Alessio. On the planning problem for a class of Mean Field Games. Comptes Rendus. Mathématique, Tome 351 (2013) no. 11-12, pp. 457-462. doi: 10.1016/j.crma.2013.07.004

[1] Achdou, Y.; Camilli, F.; Capuzzo Dolcetta, I. Mean field games: numerical methods for the planning problem, SIAM J. Control Optim., Volume 50 (2012), pp. 77-109

[2] Benamou, J.-D.; Brennier, Y. A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem, Numer. Math., Volume 84 (2000), pp. 375-393

[3] Cardaliaguet, P.; Lasry, J.-M.; Lions, P.-L.; Porretta, A. Long time average of mean field games, Netw. Heterog. Media, Volume 7 (2012), pp. 279-301

[4] Guéant, O.; Lasry, J.-M.; Lions, P.-L. Application of mean field games to growth theory, Paris–Princeton Lectures on Mathematical Finance 2010, Lect. Notes Math., Springer, Berlin, 2011

[5] Lasry, J.-M.; Lions, P.-L. Jeux à champ moyen. I. Le cas stationnaire, C. R. Acad. Sci. Paris, Ser. I, Volume 343 (2006), pp. 619-625

[6] Lasry, J.-M.; Lions, P.-L. Jeux à champ moyen. II. Horizon fini et contròle optimal, C. R. Acad. Sci. Paris, Ser. I, Volume 343 (2006), pp. 679-684

[7] Lasry, J.-M.; Lions, P.-L. Mean field games, Jpn. J. Math., Volume 2 (2007), pp. 229-260

[8] Lions, P.-L. Cours au Collège de France www.college-de-france.fr

[9] Porretta, A. On the planning problem for the Mean Field Games system, Dyn. Games Appl. (2013) (in press) | DOI

[10] A. Porretta, Weak solutions to Fokker–Planck equations and Mean Field Games, in preparation.

Cité par Sources :