On propose dans cette Note une transformation qui découple les systèmes de jeux à champ moyen stationnaires pour des hamiltoniens superlinéaires de la forme , et qui transforme l'équation de Hamilton–Jacobi–Bellman en une équation quasi linéaire introduisant le r-laplacien. Une telle transformaton nécessite une hypothèse sur la solution : cette hypothèse est satisfaite, par exemple, dans le cas unidimensionnel ou dans le cas où la solution est radiale.
In this note we propose a transformation that decouples stationary Mean-Field Games systems with superlinear Hamiltonians of the form , , and turns the Hamilton–Jacobi–Bellman equation into a quasi-linear equation involving the r-Laplace operator. Such a transformation requires an assumption on solutions to the system, which is satisfied for example in space dimension one or if solutions are radial.
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@article{CRMATH_2015__353_9_807_0, author = {Cirant, Marco}, title = {A generalization of the {Hopf{\textendash}Cole} transformation for stationary {Mean-Field} {Games} systems}, journal = {Comptes Rendus. Math\'ematique}, pages = {807--811}, publisher = {Elsevier}, volume = {353}, number = {9}, year = {2015}, doi = {10.1016/j.crma.2015.06.016}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.crma.2015.06.016/} }
TY - JOUR AU - Cirant, Marco TI - A generalization of the Hopf–Cole transformation for stationary Mean-Field Games systems JO - Comptes Rendus. Mathématique PY - 2015 SP - 807 EP - 811 VL - 353 IS - 9 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2015.06.016/ DO - 10.1016/j.crma.2015.06.016 LA - en ID - CRMATH_2015__353_9_807_0 ER -
%0 Journal Article %A Cirant, Marco %T A generalization of the Hopf–Cole transformation for stationary Mean-Field Games systems %J Comptes Rendus. Mathématique %D 2015 %P 807-811 %V 353 %N 9 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2015.06.016/ %R 10.1016/j.crma.2015.06.016 %G en %F CRMATH_2015__353_9_807_0
Cirant, Marco. A generalization of the Hopf–Cole transformation for stationary Mean-Field Games systems. Comptes Rendus. Mathématique, Tome 353 (2015) no. 9, pp. 807-811. doi : 10.1016/j.crma.2015.06.016. http://archive.numdam.org/articles/10.1016/j.crma.2015.06.016/
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