June  2012, 7(2): 337-347. doi: 10.3934/nhm.2012.7.337

A modest proposal for MFG with density constraints

1. 

Laboratoire de Mathématiques d'Orsay, Faculté de Sciences, Université Paris-Sud, 91405 Orsay cedex, France

Received  November 2011 Revised  March 2012 Published  June 2012

We consider a typical problem in Mean Field Games: the congestion case, where in the cost that agents optimize there is a penalization for passing through zones with high density of agents, in a deterministic framework. This equilibrium problem is known to be equivalent to the optimization of a global functional including an $L^p$ norm of the density. The question arises as to produce a similar model replacing the $L^p$ penalization with an $L^\infty$ constraint, but the simplest approaches do not give meaningful definitions. Taking into account recent works about crowd motion, where the density constraint $\rho\leq 1$ was treated in terms of projections of the velocity field onto the set of admissible velocity (with a constraint on the divergence) and a pressure field was introduced, we propose a definition and write a system of PDEs including the usual Hamilton-Jacobi equation coupled with the continuity equation. For this system, we analyze an example and propose some open problems.
Citation: Filippo Santambrogio. A modest proposal for MFG with density constraints. Networks & Heterogeneous Media, 2012, 7 (2) : 337-347. doi: 10.3934/nhm.2012.7.337
References:
[1]

Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5), 19 (1995), 191-246.  Google Scholar

[2]

Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005.  Google Scholar

[3]

Numer. Math., 84 (2000), 375-393. doi: 10.1007/s002110050002.  Google Scholar

[4]

SIAM J. Control Optim., 48 (2009), 1961-1976. doi: 10.1137/07070543X.  Google Scholar

[5]

Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser Boston, Inc., Boston, MA, 1993.  Google Scholar

[6]

in "Boundary Value Problems for PDE and Applications" (eds. C. Baiocchi and J. L. Lions), RMA Res. Notes Appl. Math., 29, Masson, Paris, (1993), 81-98.  Google Scholar

[7]

SIAM J. Math. Anal., 29 (1998), 1-17. doi: 10.1137/S0036141096303359.  Google Scholar

[8]

C. R. Math. Acad. Sci. Paris, 343 (2006), 679-684. doi: 10.1016/j.crma.2006.09.018.  Google Scholar

[9]

Japan. J. Math, 2 (2007), 229-260.  Google Scholar

[10]

Mat. Mod. Meth. Appl. Sci., 20 (2010), 1787-1821. doi: 10.1142/S0218202510004799.  Google Scholar

[11]

Net. Het. Media, 6 (2011), 485-519.  Google Scholar

[12]

Traffic and Granular Flow, Springer, 2007. Google Scholar

[13]

Adv. Math., 128 (1997), 153-179. doi: 10.1006/aima.1997.1634.  Google Scholar

[14]

Comm. Partial Differential Equations, 26 (2001), 101-174. doi: 10.1081/PDE-100002243.  Google Scholar

[15]

Disc. Cont. Dyn. Systems, 31 (2011), 1427-1451. doi: 10.3934/dcds.2011.31.1427.  Google Scholar

[16]

Grad. Stud. Math., 58, AMS, Providence, RI, 2003.  Google Scholar

[17]

Grundlehren der Mathematischen Wissenschaften, 338, Springer-Verlag, Berlin, 2009.  Google Scholar

show all references

References:
[1]

Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5), 19 (1995), 191-246.  Google Scholar

[2]

Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005.  Google Scholar

[3]

Numer. Math., 84 (2000), 375-393. doi: 10.1007/s002110050002.  Google Scholar

[4]

SIAM J. Control Optim., 48 (2009), 1961-1976. doi: 10.1137/07070543X.  Google Scholar

[5]

Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser Boston, Inc., Boston, MA, 1993.  Google Scholar

[6]

in "Boundary Value Problems for PDE and Applications" (eds. C. Baiocchi and J. L. Lions), RMA Res. Notes Appl. Math., 29, Masson, Paris, (1993), 81-98.  Google Scholar

[7]

SIAM J. Math. Anal., 29 (1998), 1-17. doi: 10.1137/S0036141096303359.  Google Scholar

[8]

C. R. Math. Acad. Sci. Paris, 343 (2006), 679-684. doi: 10.1016/j.crma.2006.09.018.  Google Scholar

[9]

Japan. J. Math, 2 (2007), 229-260.  Google Scholar

[10]

Mat. Mod. Meth. Appl. Sci., 20 (2010), 1787-1821. doi: 10.1142/S0218202510004799.  Google Scholar

[11]

Net. Het. Media, 6 (2011), 485-519.  Google Scholar

[12]

Traffic and Granular Flow, Springer, 2007. Google Scholar

[13]

Adv. Math., 128 (1997), 153-179. doi: 10.1006/aima.1997.1634.  Google Scholar

[14]

Comm. Partial Differential Equations, 26 (2001), 101-174. doi: 10.1081/PDE-100002243.  Google Scholar

[15]

Disc. Cont. Dyn. Systems, 31 (2011), 1427-1451. doi: 10.3934/dcds.2011.31.1427.  Google Scholar

[16]

Grad. Stud. Math., 58, AMS, Providence, RI, 2003.  Google Scholar

[17]

Grundlehren der Mathematischen Wissenschaften, 338, Springer-Verlag, Berlin, 2009.  Google Scholar

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