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On the Planning Problem for the Mean Field Games System

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  • Alessio Porretta

Abstract

We consider the planning problem for a class of mean field games, consisting in a coupled system of a Hamilton–Jacobi–Bellman equation for the value function u and a Fokker–Planck equation for the density m of the players, whereas one wishes to drive the density of players from the given initial configuration to a target one at time T through the optimal decisions of the agents. Assuming that the coupling F(x,m) in the cost criterion is monotone with respect to m, and that the Hamiltonian has some growth bounded below and above by quadratic functions, we prove the existence of a weak solution to the system with prescribed initial and terminal conditions m 0 , m 1 (positive and smooth) for the density m. This is also a special case of an exact controllability result for the Fokker–Planck equation through some optimal transport field. Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • Alessio Porretta, 2014. "On the Planning Problem for the Mean Field Games System," Dynamic Games and Applications, Springer, vol. 4(2), pages 231-256, June.
    • Handle: RePEc:spr:dyngam:v:4:y:2014:i:2:p:231-256
    DOI: 10.1007/s13235-013-0080-0
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    Cited by:

    1. Yongxin Chen & Tryphon T. Georgiou & Michele Pavon, 2018. "Steering the Distribution of Agents in Mean-Field Games System," Journal of Optimization Theory and Applications, Springer, vol. 179(1), pages 332-357, October.
    2. Arthur Fleig & Roberto Guglielmi, 2017. "Optimal Control of the Fokker–Planck Equation with Space-Dependent Controls," Journal of Optimization Theory and Applications, Springer, vol. 174(2), pages 408-427, August.
    3. Diogo A. Gomes & Levon Nurbekyan & Mariana Prazeres, 2018. "One-Dimensional Stationary Mean-Field Games with Local Coupling," Dynamic Games and Applications, Springer, vol. 8(2), pages 315-351, June.
    4. Noha Almulla & Rita Ferreira & Diogo Gomes, 2017. "Two Numerical Approaches to Stationary Mean-Field Games," Dynamic Games and Applications, Springer, vol. 7(4), pages 657-682, December.

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