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Numerical Solution of Mean Field Games Problems with Turnpike Effect


We present a problem described by Mean Field Games (MFG) and Optimal Control theory on finite time horizon. This problem consists of a system of PDEs: a Kolmogorov–Fokker–Planck equation, evolving forward in time and a Hamilton–Jacobi–Bellman equation, evolving backwards in time. The numerical difficulties are based on a turnpike effect considered in this paper. We present an extremal problem whose necessary conditions of extremal satisfy the initial system of PDEs, and introduce its numerical solution at the heart of monotonic schemes. According to special assumptions, PDEs can be reduced to Riccati ODEs. We consider this reduction as a test example for the numerical solution of the extremal problem.

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The author would like to express his sincere gratitude to scientific leader, corresponding member of the Russian Academy of Sciences, Doctor of Physical and Mathematical sciences, A. A. Shananin for the guidance and help with Mean Field Games and Optimal Control theory.


The work has been supported by RSF (grant 16-11-10246).

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Correspondence to N. V. Trusov.

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(Submitted by A. V. Lapin)

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Trusov, N.V. Numerical Solution of Mean Field Games Problems with Turnpike Effect. Lobachevskii J Math 41, 561–576 (2020).

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Keywords and phrases:

  • mean field games
  • optimal control
  • turnpike effect
  • numerical solution
  • monotonic schemes