A semi-discrete  approximation for a  first order mean field  game problem

Fabio Camilli; Francisco Silva; Fabio Camilli; Francisco Silva

doi:10.3934/nhm.2012.7.263

Networks and Heterogeneous Media

2012, Volume 7, Issue 2: 263-277. doi: 10.3934/nhm.2012.7.263

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A semi-discrete approximation for a first order mean field game problem

Fabio Camilli ^{1
,},
Francisco Silva ^{2
,}

1.
"Sapienza", Università di Roma, Dipartimento di Scienze di Base e Applicate per l'Ingegneria, 00161 Roma
2.
"Sapienza", Università di Roma, Dipartimento di Matematica Guido Castelnuovo, 00185 Rome

Received: 01 November 2011 Revised: 01 March 2012
Primary: 91A13; Secondary: 65M25, 49L25.

In this article we consider a model first order mean field game problem, introduced by J.M. Lasry and P.L. Lions in [18]. Its solution $(v,m)$ can be obtained as the limit of the solutions of the second order mean field game problems, when the noise parameter tends to zero (see [18]). We propose a semi-discrete in time approximation of the system and, under natural assumptions, we prove that it is well posed and that it converges to $(v,m)$ when the discretization parameter tends to zero.
- Mean field games,
- optimal control,
- convex duality,
- numerical methods
Citation: Fabio Camilli, Francisco Silva. A semi-discrete approximation for a first order mean field game problem[J]. Networks and Heterogeneous Media, 2012, 7(2): 263-277. doi: 10.3934/nhm.2012.7.263

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Abstract

In this article we consider a model first order mean field game problem, introduced by J.M. Lasry and P.L. Lions in [18]. Its solution $(v,m)$ can be obtained as the limit of the solutions of the second order mean field game problems, when the noise parameter tends to zero (see [18]). We propose a semi-discrete in time approximation of the system and, under natural assumptions, we prove that it is well posed and that it converges to $(v,m)$ when the discretization parameter tends to zero.

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