Nonlinear elliptic systems and mean-field games
We consider a class of quasilinear elliptic systems of PDEs consisting of N Hamilton–Jacobi–Bellman equations coupled with N divergence form equations, generalising to N > 1 populations the PDEs for stationary Mean-Field Games first proposed by Lasry and Lions. We provide a wide range of suffici...
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| Veröffentlicht in: | Nonlinear differential equations and applications 2016-08, Vol.23 (4), Article 44 |
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| creator | Bardi, Martino Feleqi, Ermal |
| description | We consider a class of quasilinear elliptic systems of PDEs consisting of
N
Hamilton–Jacobi–Bellman equations coupled with
N
divergence form equations, generalising to
N
> 1 populations the PDEs for stationary Mean-Field Games first proposed by Lasry and Lions. We provide a wide range of sufficient conditions for the existence of solutions to these systems: either the Hamiltonians are required to behave at most linearly for large gradients, as it occurs when the controls of the agents are bounded, or they must grow faster than linearly and not oscillate too much in the space variables, in a suitable sense. We show the connection of these systems with the classical strongly coupled systems of Hamilton–Jacobi–Bellman equations of the theory of
N
-person stochastic differential games studied by Bensoussan and Frehse. We also prove the existence of Nash equilibria in feedback form for some
N
-person games. |
| doi_str_mv | 10.1007/s00030-016-0397-7 |
| format | Article |
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N
Hamilton–Jacobi–Bellman equations coupled with
N
divergence form equations, generalising to
N
> 1 populations the PDEs for stationary Mean-Field Games first proposed by Lasry and Lions. We provide a wide range of sufficient conditions for the existence of solutions to these systems: either the Hamiltonians are required to behave at most linearly for large gradients, as it occurs when the controls of the agents are bounded, or they must grow faster than linearly and not oscillate too much in the space variables, in a suitable sense. We show the connection of these systems with the classical strongly coupled systems of Hamilton–Jacobi–Bellman equations of the theory of
N
-person stochastic differential games studied by Bensoussan and Frehse. We also prove the existence of Nash equilibria in feedback form for some
N
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N
Hamilton–Jacobi–Bellman equations coupled with
N
divergence form equations, generalising to
N
> 1 populations the PDEs for stationary Mean-Field Games first proposed by Lasry and Lions. We provide a wide range of sufficient conditions for the existence of solutions to these systems: either the Hamiltonians are required to behave at most linearly for large gradients, as it occurs when the controls of the agents are bounded, or they must grow faster than linearly and not oscillate too much in the space variables, in a suitable sense. We show the connection of these systems with the classical strongly coupled systems of Hamilton–Jacobi–Bellman equations of the theory of
N
-person stochastic differential games studied by Bensoussan and Frehse. We also prove the existence of Nash equilibria in feedback form for some
N
-person games.</description><subject>Analysis</subject><subject>Differential equations</subject><subject>Differential games</subject><subject>Divergence</subject><subject>Economic models</subject><subject>Game theory</subject><subject>Mathematical analysis</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Nonlinear systems</subject><issn>1021-9722</issn><issn>1420-9004</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2016</creationdate><recordtype>article</recordtype><recordid>eNp1kDtPwzAUhS0EEqXwA9giMZteP-LHiCqgSBUsMFuOc12lyqPY6dB_T6owsDDdM5zvXOkj5J7BIwPQqwwAAigwRUFYTfUFWTDJgVoAeTll4Ixazfk1ucl5D8C0EnZBVu9D3zY9-lRg2zaHsQlFPuURu1z4vi469D2NDbZ1sfMd5ltyFX2b8e73LsnXy_PnekO3H69v66ctDaK0I1W1gErXvpQheGYwVlwbGxQobTGUWkQtjVFCc8WZZtzUEYUswVZRiUp6sSQP8-4hDd9HzKPbD8fUTy8dMwaM4UKaqcXmVkhDzgmjO6Sm8-nkGLizFzd7cZMXd_bi9MTwmclTt99h-rP8L_QD_Y1jbA</recordid><startdate>20160801</startdate><enddate>20160801</enddate><creator>Bardi, Martino</creator><creator>Feleqi, Ermal</creator><general>Springer International Publishing</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20160801</creationdate><title>Nonlinear elliptic systems and mean-field games</title><author>Bardi, Martino ; Feleqi, Ermal</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c359t-6d30b7da54cca18efb2789c60679ec573f748863726217128dfe34509bf63b4a3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2016</creationdate><topic>Analysis</topic><topic>Differential equations</topic><topic>Differential games</topic><topic>Divergence</topic><topic>Economic models</topic><topic>Game theory</topic><topic>Mathematical analysis</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Nonlinear systems</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Bardi, Martino</creatorcontrib><creatorcontrib>Feleqi, Ermal</creatorcontrib><collection>CrossRef</collection><jtitle>Nonlinear differential equations and applications</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Bardi, Martino</au><au>Feleqi, Ermal</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Nonlinear elliptic systems and mean-field games</atitle><jtitle>Nonlinear differential equations and applications</jtitle><stitle>Nonlinear Differ. Equ. Appl</stitle><date>2016-08-01</date><risdate>2016</risdate><volume>23</volume><issue>4</issue><artnum>44</artnum><issn>1021-9722</issn><eissn>1420-9004</eissn><abstract>We consider a class of quasilinear elliptic systems of PDEs consisting of
N
Hamilton–Jacobi–Bellman equations coupled with
N
divergence form equations, generalising to
N
> 1 populations the PDEs for stationary Mean-Field Games first proposed by Lasry and Lions. We provide a wide range of sufficient conditions for the existence of solutions to these systems: either the Hamiltonians are required to behave at most linearly for large gradients, as it occurs when the controls of the agents are bounded, or they must grow faster than linearly and not oscillate too much in the space variables, in a suitable sense. We show the connection of these systems with the classical strongly coupled systems of Hamilton–Jacobi–Bellman equations of the theory of
N
-person stochastic differential games studied by Bensoussan and Frehse. We also prove the existence of Nash equilibria in feedback form for some
N
-person games.</abstract><cop>Cham</cop><pub>Springer International Publishing</pub><doi>10.1007/s00030-016-0397-7</doi><oa>free_for_read</oa></addata></record> |
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| ispartof | Nonlinear differential equations and applications, 2016-08, Vol.23 (4), Article 44 |
| issn | 1021-9722 1420-9004 |
| language | eng |
| recordid | cdi_proquest_journals_1880882348 |
| source | SpringerLink Journals - AutoHoldings |
| subjects | Analysis Differential equations Differential games Divergence Economic models Game theory Mathematical analysis Mathematics Mathematics and Statistics Nonlinear systems |
| title | Nonlinear elliptic systems and mean-field games |
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