Time-dependent mean-field games in the superquadratic case

Handle URI:
http://hdl.handle.net/10754/608650
Title:
Time-dependent mean-field games in the superquadratic case
Authors:
Gomes, Diogo A. ( 0000-0002-3129-3956 ) ; Pimentel, Edgard; Sánchez-Morgado, Héctor
Abstract:
We investigate time-dependent mean-field games with superquadratic Hamiltonians and a power dependence on the measure. Such problems pose substantial mathematical challenges as key techniques used in the subquadratic case, which was studied in a previous publication of the authors, do not extend to the superquadratic setting. The main objective of the present paper is to address these difficulties. Because of the superquadratic structure of the Hamiltonian, Lipschitz estimates for the solutions of the Hamilton−Jacobi equation are obtained here through a novel set of techniques. These explore the parabolic nature of the problem through the nonlinear adjoint method. Well-posedness is proven by combining Lipschitz regularity for the Hamilton−Jacobi equation with polynomial estimates for solutions of the Fokker−Planck equation. Existence of classical solutions is then established under conditions depending only on the growth of the Hamiltonian and the dimension. Our results also add to current understanding of superquadratic Hamilton−Jacobi equations.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division; Center for Uncertainty Quantification in Computational Science and Engineering (SRI-UQ)
Citation:
Time-dependent mean-field games in the superquadratic case 2016, 22 (2):562 ESAIM: Control, Optimisation and Calculus of Variations
Publisher:
EDP Sciences
Journal:
ESAIM: Control, Optimisation and Calculus of Variations
Issue Date:
6-Apr-2016
DOI:
10.1051/cocv/2015029
Type:
Article
ISSN:
1292-8119; 1262-3377
Sponsors:
D. Gomes was partially supported by KAUST baseline funds, KAUST SRI, Center for Uncertainty Quantification in Computational Science and Engineering. E. Pimentel was supported by CNPq-Brazil. The authors thank Cardaliaguet, Lions, Porretta and Souganidis for very useful comments and suggestions.
Additional Links:
http://www.esaim-cocv.org/10.1051/cocv/2015029
Appears in Collections:
Articles; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorGomes, Diogo A.en
dc.contributor.authorPimentel, Edgarden
dc.contributor.authorSánchez-Morgado, Héctoren
dc.date.accessioned2016-05-09T08:18:16Zen
dc.date.available2016-05-09T08:18:16Zen
dc.date.issued2016-04-06en
dc.identifier.citationTime-dependent mean-field games in the superquadratic case 2016, 22 (2):562 ESAIM: Control, Optimisation and Calculus of Variationsen
dc.identifier.issn1292-8119en
dc.identifier.issn1262-3377en
dc.identifier.doi10.1051/cocv/2015029en
dc.identifier.urihttp://hdl.handle.net/10754/608650en
dc.description.abstractWe investigate time-dependent mean-field games with superquadratic Hamiltonians and a power dependence on the measure. Such problems pose substantial mathematical challenges as key techniques used in the subquadratic case, which was studied in a previous publication of the authors, do not extend to the superquadratic setting. The main objective of the present paper is to address these difficulties. Because of the superquadratic structure of the Hamiltonian, Lipschitz estimates for the solutions of the Hamilton−Jacobi equation are obtained here through a novel set of techniques. These explore the parabolic nature of the problem through the nonlinear adjoint method. Well-posedness is proven by combining Lipschitz regularity for the Hamilton−Jacobi equation with polynomial estimates for solutions of the Fokker−Planck equation. Existence of classical solutions is then established under conditions depending only on the growth of the Hamiltonian and the dimension. Our results also add to current understanding of superquadratic Hamilton−Jacobi equations.en
dc.description.sponsorshipD. Gomes was partially supported by KAUST baseline funds, KAUST SRI, Center for Uncertainty Quantification in Computational Science and Engineering. E. Pimentel was supported by CNPq-Brazil. The authors thank Cardaliaguet, Lions, Porretta and Souganidis for very useful comments and suggestions.en
dc.language.isoenen
dc.publisherEDP Sciencesen
dc.relation.urlhttp://www.esaim-cocv.org/10.1051/cocv/2015029en
dc.rightsArchived with thanks to ESAIM: Control, Optimisation and Calculus of Variationsen
dc.subjectMean-field gamesen
dc.subjectinitial terminal value problemen
dc.subjectsuperquadratic Hamiltoniansen
dc.subjectnonlinear adjoint methoden
dc.titleTime-dependent mean-field games in the superquadratic caseen
dc.typeArticleen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.contributor.departmentCenter for Uncertainty Quantification in Computational Science and Engineering (SRI-UQ)en
dc.identifier.journalESAIM: Control, Optimisation and Calculus of Variationsen
dc.eprint.versionPublisher's Version/PDFen
dc.contributor.institutionUniversidade Federal do São Carlos, Department of Mathematics, 13560-250 São Carlos-SP, Brazilen
dc.contributor.institutionInstituto de Matemáticas, Universidad Nacional Autónoma de México. DF 04510 México, Méxicoen
dc.contributor.affiliationKing Abdullah University of Science and Technology (KAUST)en
kaust.authorGomes, Diogo A.en
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