{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,20]],"date-time":"2025-05-20T04:01:29Z","timestamp":1747713689815,"version":"3.40.5"},"reference-count":48,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2024,10,6]],"date-time":"2024-10-06T00:00:00Z","timestamp":1728172800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,10,6]],"date-time":"2024-10-06T00:00:00Z","timestamp":1728172800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"name":"MIUR Excellence Department Project","award":["E83C23000330006","E83C23000330006"],"award-info":[{"award-number":["E83C23000330006","E83C23000330006"]}]},{"name":"INdAM-GNAMPA","award":["E53C22001930001","E53C22001930001"],"award-info":[{"award-number":["E53C22001930001","E53C22001930001"]}]},{"DOI":"10.13039\/501100004052","name":"King Abdullah University of Science and Technology","doi-asserted-by":"publisher","award":["CRG2021-4674","CRG2021-4674"],"award-info":[{"award-number":["CRG2021-4674","CRG2021-4674"]}],"id":[{"id":"10.13039\/501100004052","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Dyn Games Appl"],"published-print":{"date-parts":[[2025,5]]},"DOI":"10.1007\/s13235-024-00594-3","type":"journal-article","created":{"date-parts":[[2024,10,6]],"date-time":"2024-10-06T12:01:32Z","timestamp":1728216092000},"page":"592-609","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Rate of Convergence for First-Order Singular Perturbation Problems: Hamilton\u2013Jacobi\u2013Isaacs Equations and Mean Field Games of Acceleration"],"prefix":"10.1007","volume":"15","author":[{"given":"Piermarco","family":"Cannarsa","sequence":"first","affiliation":[]},{"given":"Cristian","family":"Mendico","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,10,6]]},"reference":[{"issue":"3","key":"594_CR1","doi-asserted-by":"publisher","first-page":"32","DOI":"10.1007\/s00030-020-00634-y","volume":"27","author":"Y Achdou","year":"2020","unstructured":"Achdou Y, Mannucci P, Marchi C, Tchou N (2020) Deterministic mean field games with control on the acceleration. NoDEA Nonlinear Differ Equ Appl 27(3):32","journal-title":"NoDEA Nonlinear Differ Equ Appl"},{"issue":"1","key":"594_CR2","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1007\/s00205-003-0266-5","volume":"170","author":"O Alvarez","year":"2003","unstructured":"Alvarez O, Bardi M (2003) Singular perturbations of nonlinear degenerate parabolic PDEs: a general convergence result. Arch Ration Mech Anal 170(1):17\u201361","journal-title":"Arch Ration Mech Anal"},{"key":"594_CR3","doi-asserted-by":"crossref","unstructured":"Alvarez O, Bardi M (2010) Ergodicity, stabilization, and singular perturbations for Bellman\u2013Isaacs equations, vol 960. American Mathematical Society (AMS), Providence, RI","DOI":"10.1090\/S0065-9266-09-00588-2"},{"issue":"5\u20136","key":"594_CR4","doi-asserted-by":"publisher","first-page":"621","DOI":"10.1016\/S0362-546X(97)00427-6","volume":"31","author":"L Amour","year":"1998","unstructured":"Amour L, Ben-Artzi M (1998) Global existence and decay for viscous Hamilton\u2013Jacobi equations. Nonlinear Anal Theory Methods Appl 31(5\u20136):621\u2013628","journal-title":"Nonlinear Anal Theory Methods Appl"},{"key":"594_CR5","doi-asserted-by":"publisher","DOI":"10.1016\/j.na.2020.112199","volume":"204","author":"M Bardi","year":"2021","unstructured":"Bardi M, Cardaliaguet P (2021) Convergence of some mean field games systems to aggregation and flocking models. Nonlinear Anal 204:112199","journal-title":"Nonlinear Anal"},{"issue":"1\u20132","key":"594_CR6","first-page":"1","volume":"112","author":"G Barles","year":"2019","unstructured":"Barles G, Ley O, Nguyen T-T, Phan TV (2019) Large time behavior of unbounded solutions of first-order Hamilton-Jacobi equations in $$\\mathbb{R}^{N}$$. Asymptot Anal 112(1\u20132):1\u201322","journal-title":"Asymptot Anal"},{"key":"594_CR7","unstructured":"Bensoussan A(1988) Perturbation methods in optimal control. Transl. from the French by C. Tomson. Gauthier-Villars, Paris; Wiley, Chichester"},{"key":"594_CR8","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4614-8508-7","volume-title":"Mean field games and mean field type control theory","author":"A Bensoussan","year":"2013","unstructured":"Bensoussan A, Frehse J, Yam P (2013) Mean field games and mean field type control theory. Springer, New York"},{"key":"594_CR9","doi-asserted-by":"crossref","unstructured":"Camilli F, Goffi A, Mendico C (2023) Quantitative and qualitative properties for Hamilton\u2013Jacobi PDEs via the nonlinear adjoint method. arXiv:2307.12932","DOI":"10.2422\/2036-2145.202404_015"},{"issue":"2","key":"594_CR10","doi-asserted-by":"publisher","first-page":"405","DOI":"10.1515\/ans-2011-0210","volume":"11","author":"F Camilli","year":"2011","unstructured":"Camilli F, Cesaroni A, Marchi C (2011) Homogenization and vanishing viscosity in fully nonlinear elliptic equations: rate of convergence estimates. Adv Nonlinear Stud 11(2):405\u2013428","journal-title":"Adv Nonlinear Stud"},{"issue":"1","key":"594_CR11","doi-asserted-by":"publisher","first-page":"61","DOI":"10.3934\/nhm.2011.6.61","volume":"6","author":"F Camilli","year":"2011","unstructured":"Camilli F, Marchi C (2011) On the convergence rate in multiscale homogenization of fully nonlinear elliptic problems. Netw Heterog Med 6(1):61\u201375","journal-title":"Netw Heterog Med"},{"issue":"2","key":"594_CR12","first-page":"221","volume":"5","author":"P Cannarsa","year":"2020","unstructured":"Cannarsa P, Mendico C (2020) Mild and weak solutions of mean field game problems for linear control systems. Minimax Theory Appl 5(2):221\u2013250","journal-title":"Minimax Theory Appl"},{"issue":"3","key":"594_CR13","doi-asserted-by":"publisher","first-page":"1113","DOI":"10.1512\/iumj.2001.50.1933","volume":"50","author":"I Capuzzo-Dolcetta","year":"2001","unstructured":"Capuzzo-Dolcetta I, Ishii H (2001) On the rate of convergence in homogenization of Hamilton\u2013Jacobi equations. Indiana Univ Math J 50(3):1113\u20131129","journal-title":"Indiana Univ Math J"},{"key":"594_CR14","unstructured":"Cardaliaguet P(2012) Notes on mean field games from P.-L. Lions lectures at Coll\u00e9ge de France. Unpublished"},{"key":"594_CR15","doi-asserted-by":"crossref","unstructured":"Carmona R, Delarue F(2018) Probabilistic theory of mean field games with applications I. Mean field FBSDEs, control, and games, vol\u00a083. Springer, Cham","DOI":"10.1007\/978-3-319-56436-4"},{"issue":"4","key":"594_CR16","doi-asserted-by":"publisher","first-page":"2701","DOI":"10.1137\/16M1063459","volume":"48","author":"A Cesaroni","year":"2016","unstructured":"Cesaroni A, Dirr N, Marchi C (2016) Homogenization of a mean field game system in the small noise limit. SIAM J Math Anal 48(4):2701\u20132729","journal-title":"SIAM J Math Anal"},{"issue":"6","key":"594_CR17","doi-asserted-by":"publisher","first-page":"4001","DOI":"10.1137\/140953903","volume":"52","author":"M Cirant","year":"2015","unstructured":"Cirant M (2015) On the solvability of some ergodic control problems in $$\\mathbb{R} ^{d}$$. SIAM J Control Optim 52(6):4001\u20134026","journal-title":"SIAM J Control Optim"},{"issue":"4","key":"594_CR18","doi-asserted-by":"publisher","first-page":"757","DOI":"10.1016\/j.anihpc.2020.01.006","volume":"37","author":"M Cirant","year":"2020","unstructured":"Cirant M, Goffi A (2020) Lipschitz regularity for viscous Hamilton-Jacobi equations with $$L^p$$ terms. Ann Inst Henri Poincar\u00e9 Anal Non Lin\u00e9aire 37(4):757\u2013784","journal-title":"Ann Inst Henri Poincar\u00e9 Anal Non Lin\u00e9aire"},{"issue":"1","key":"594_CR19","doi-asserted-by":"publisher","first-page":"23","DOI":"10.1137\/18M1179328","volume":"57","author":"E Chasseigne","year":"2019","unstructured":"Chasseigne E, Ichihara N (2019) Ergodic problems for viscous Hamilton\u2013Jacobi equations with inward drift. SIAM J Control Optim 57(1):23\u201352","journal-title":"SIAM J Control Optim"},{"issue":"1","key":"594_CR20","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1137\/100797515","volume":"9","author":"E Cristiani","year":"2011","unstructured":"Cristiani E, Piccoli B, Tosin A (2011) Multiscale modeling of granular flows with application to crowd dynamics. Multiscale Model Simul 9(1):155\u2013182","journal-title":"Multiscale Model Simul"},{"key":"594_CR21","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-06620-2","volume-title":"Multiscale modeling of pedestrian dynamics","author":"E Cristiani","year":"2014","unstructured":"Cristiani E, Piccoli B, Tosin A (2014) Multiscale modeling of pedestrian dynamics. Springer, Cham"},{"issue":"2","key":"594_CR22","doi-asserted-by":"publisher","first-page":"159","DOI":"10.1007\/s00526-002-0164-y","volume":"17","author":"LC Evans","year":"2003","unstructured":"Evans LC (2003) Some new PDE methods for weak KAM theory. Calc Var Partial Differ Equ 17(2):159\u2013177","journal-title":"Calc Var Partial Differ Equ"},{"key":"594_CR23","doi-asserted-by":"publisher","first-page":"773","DOI":"10.1512\/iumj.1984.33.33040","volume":"33","author":"LC Evans","year":"1984","unstructured":"Evans LC, Souganidis PE (1984) Differential games and representation formulas for solutions of Hamilton\u2013Jacobi\u2013Isaacs equations. Indiana Univ Math J 33:773\u2013797","journal-title":"Indiana Univ Math J"},{"issue":"1\u20132","key":"594_CR24","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00030-007-2047-6","volume":"14","author":"A Fathi","year":"2007","unstructured":"Fathi A, Maderna E (2007) Weak KAM theorem on non compact manifolds. NoDEA Nonlinear Differ Equ Appl 14(1\u20132):1\u201327","journal-title":"NoDEA Nonlinear Differ Equ Appl"},{"issue":"2","key":"594_CR25","doi-asserted-by":"publisher","first-page":"293","DOI":"10.1512\/iumj.1989.38.38015","volume":"38","author":"WH Fleming","year":"1989","unstructured":"Fleming WH, Souganidis PE (1989) On the existence of value functions of two-player, zero-sum stochastic differential games. Indiana Univ Math J 38(2):293\u2013314","journal-title":"Indiana Univ Math J"},{"issue":"2","key":"594_CR26","doi-asserted-by":"publisher","first-page":"605","DOI":"10.1051\/cocv\/2017051","volume":"24","author":"D Ghilli","year":"2018","unstructured":"Ghilli D (2018) Viscosity methods for large deviations estimates of multiscale stochastic processes. ESAIM Control Optim Calc Var 24(2):605\u2013637","journal-title":"ESAIM Control Optim Calc Var"},{"issue":"1","key":"594_CR27","doi-asserted-by":"publisher","first-page":"27","DOI":"10.1016\/j.jmaa.2023.127225","volume":"526","author":"D Ghilli","year":"2023","unstructured":"Ghilli D, Marchi C (2023) Rate of convergence for singular perturbations of Hamilton\u2013Jacobi equations in unbounded spaces. J Math Anal Appl 526(1):27","journal-title":"J Math Anal Appl"},{"key":"594_CR28","unstructured":"Giannetti J, Silva FJ(2022) Approximation of deterministic mean field games with control-affine dynamics. arXiv:2212.09854"},{"key":"594_CR29","unstructured":"Giannetti J, Silva FJ, Zorkot A(2023) Approximation of deterministic mean field games under polynomial growth conditions on the data. arXiv:2305.01445"},{"key":"594_CR30","doi-asserted-by":"crossref","unstructured":"Gomes DA, Pimentel EA, Voskanyan V (2016) Regularity theory for mean-field game systems. Springer, Cham; Rio de Janeiro: Sociedade Brasileira de Matem\u00e1tica Aplicada e Computacional (SBMAC)","DOI":"10.1007\/978-3-319-38934-9"},{"issue":"3","key":"594_CR31","doi-asserted-by":"publisher","first-page":"221","DOI":"10.4310\/CIS.2006.v6.n3.a5","volume":"6","author":"M Huang","year":"2006","unstructured":"Huang M, Malham\u00e9 RP, Caines PE (2006) Large population stochastic dynamic games: closed-loop McKean\u2013Vlasov systems and the Nash certainty equivalence principle. Commun Inf Syst 6(3):221\u2013252","journal-title":"Commun Inf Syst"},{"issue":"9","key":"594_CR32","doi-asserted-by":"publisher","first-page":"1560","DOI":"10.1109\/TAC.2007.904450","volume":"52","author":"M Huang","year":"2007","unstructured":"Huang M, Malham\u00e9 RP, Caines PE (2007) Large-population cost-coupled LQG problems with nonuniform agents: individual-mass behavior and decentralized $$\\epsilon $$-Nash equilibria. IEEE Trans Autom Control 52(9):1560\u20131571","journal-title":"IEEE Trans Autom Control"},{"key":"594_CR33","doi-asserted-by":"publisher","first-page":"383","DOI":"10.1007\/s00205-008-0170-0","volume":"194","author":"N Ichihara","year":"2009","unstructured":"Ichihara N, Ishii H (2009) Long-time behavior of solutions of Hamilton-Jacobi equations with convex and coercive Hamiltonians. Arch Ration Mech Anal 194:383\u2013419","journal-title":"Arch Ration Mech Anal"},{"issue":"2","key":"594_CR34","doi-asserted-by":"publisher","first-page":"231","DOI":"10.1016\/j.anihpc.2006.09.002","volume":"25","author":"H Ishii","year":"2008","unstructured":"Ishii H (2008) Asymptotic solutions for large time of Hamilton\u2013Jacobi equations in Euclidean n space. Ann Inst Henri Poincar\u00e9 Anal Non Lin\u00e9aire 25(2):231\u2013266","journal-title":"Ann Inst Henri Poincar\u00e9 Anal Non Lin\u00e9aire"},{"issue":"4","key":"594_CR35","doi-asserted-by":"publisher","first-page":"709","DOI":"10.1137\/130925669","volume":"55","author":"AJ Bernoff","year":"2013","unstructured":"Bernoff AJ, Topaz CM (2013) Nonlocal aggregation models: a primer of swarm equilibria. SIAM Rev 55(4):709\u2013747","journal-title":"SIAM Rev"},{"key":"594_CR36","unstructured":"Kokotovi\u0107 PV, Khalil HK, O\u2019Reilly J (1986) Singular perturbation methods in control: analysis and design. Academic Press, London etc. (Harcourt Brace Jovanovich, Publishers). XII"},{"issue":"9","key":"594_CR37","doi-asserted-by":"publisher","first-page":"619","DOI":"10.1016\/j.crma.2006.09.019","volume":"343","author":"J-M Lasry","year":"2006","unstructured":"Lasry J-M, Lions P-L (2006) Jeux \u00e0 champ moyen. I: Le cas stationnaire. C R Math Acad Sci Paris 343(9):619\u2013625","journal-title":"C R Math Acad Sci Paris"},{"issue":"10","key":"594_CR38","doi-asserted-by":"publisher","first-page":"679","DOI":"10.1016\/j.crma.2006.09.018","volume":"343","author":"J-M Lasry","year":"2006","unstructured":"Lasry J-M, Lions P-L (2006) Jeux \u00e0 champ moyen. II: Horizon fini et contr\u00f4le optimal. C R Math Acad Sci Paris 343(10):679\u2013684","journal-title":"C R Math Acad Sci Paris"},{"issue":"1","key":"594_CR39","doi-asserted-by":"publisher","first-page":"229","DOI":"10.1007\/s11537-007-0657-8","volume":"2","author":"J-M Lasry","year":"2007","unstructured":"Lasry J-M, Lions P-L (2007) Mean field games. Jpn J Math (3) 2(1):229\u2013260","journal-title":"Jpn J Math (3)"},{"key":"594_CR40","unstructured":"Lions P-L, Papanicolaou G, Varadhan S (1986) Homogenization of Hamilton\u2013Jacobi equations. Unpublished"},{"issue":"4","key":"594_CR41","doi-asserted-by":"publisher","first-page":"733","DOI":"10.4171\/rlm\/912","volume":"31","author":"P-L Lions","year":"2020","unstructured":"Lions P-L, Souganidis PE (2020) Homogenization of the backward\u2013forward mean field games systems in periodic environments. Atti Accad Naz Lincei Cl Sci Fis Mat Nat, IX Ser, Rend Lincei Mat Appl 31(4):733\u2013755","journal-title":"Atti Accad Naz Lincei Cl Sci Fis Mat Nat, IX Ser, Rend Lincei Mat Appl"},{"issue":"4","key":"594_CR42","doi-asserted-by":"publisher","first-page":"1429","DOI":"10.1051\/cocv\/2017063","volume":"24","author":"P Mannucci","year":"2018","unstructured":"Mannucci P, Marchi C, Tchou N (2018) Singular perturbations for a subelliptic operator. ESAIM Control Optim Calc Var 24(4):1429\u20131451","journal-title":"ESAIM Control Optim Calc Var"},{"issue":"3","key":"594_CR43","doi-asserted-by":"publisher","first-page":"27","DOI":"10.1007\/s00028-023-00905-y","volume":"23","author":"C Mendico","year":"2023","unstructured":"Mendico C (2023) A singular perturbation problem for mean field games of acceleration: application to mean field games of control. J Evol Equ 23(3):27","journal-title":"J Evol Equ"},{"key":"594_CR44","unstructured":"O\u2019Malley REJ (1974) Introduction to singular perturbations. In: Applied mathematics and mechanics, vol 14. Academic Press, New York, London, a subsidiary of Harcourt Brace Jovanovich, Publishers. VIII"},{"issue":"7","key":"594_CR45","doi-asserted-by":"publisher","first-page":"1601","DOI":"10.1007\/s11538-006-9088-6","volume":"68","author":"CM Topaz","year":"2006","unstructured":"Topaz CM, Bertozzi AL, Lewis MA (2006) A nonlocal continuum model for biological aggregation. Bull Math Biol 68(7):1601\u20131623","journal-title":"Bull Math Biol"},{"issue":"4\u20136","key":"594_CR46","doi-asserted-by":"publisher","first-page":"827","DOI":"10.1080\/03605300500358087","volume":"31","author":"Y Fujita","year":"2006","unstructured":"Fujita Y, Ishii H, Loreti P (2006) Asymptotic solutions of viscous Hamilton\u2013Jacobi equations with Ornstein\u2013Uhlenbeck operator. Commun Partial Differ Equ 31(4\u20136):827\u2013848","journal-title":"Commun Partial Differ Equ"},{"key":"594_CR47","unstructured":"Tang W, Zhang YP (2023) The convergence rate of vanishing viscosity approximation for mean field games. arXiv:2303.14560"},{"key":"594_CR48","unstructured":"Yin GG, Zhang Q (1998) Continuous-time Markov chains and applications. A singular perturbation approach, volume\u00a037 of Applied Mathematics (NY). Springer, New York, NY"}],"container-title":["Dynamic Games and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s13235-024-00594-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s13235-024-00594-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s13235-024-00594-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,19]],"date-time":"2025-05-19T07:10:52Z","timestamp":1747638652000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s13235-024-00594-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,10,6]]},"references-count":48,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2025,5]]}},"alternative-id":["594"],"URL":"https:\/\/doi.org\/10.1007\/s13235-024-00594-3","relation":{},"ISSN":["2153-0785","2153-0793"],"issn-type":[{"type":"print","value":"2153-0785"},{"type":"electronic","value":"2153-0793"}],"subject":[],"published":{"date-parts":[[2024,10,6]]},"assertion":[{"value":"4 September 2024","order":1,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 October 2024","order":2,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"Not applicable.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical approval"}},{"value":"The authors report there are no Conflict of interest to declare.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}