## Abstract

In the context of mean field games, with possible control of the diffusion coefficient, we consider a path-dependent version of the planning problem introduced by P. L. Lions: given a pair of marginal distributions (\mu 0,\mu 1), find a specification of the game problem starting from the initial distribution \mu 0 and inducing the target distribution \mu 1 at the mean field game equilibrium. Our main result reduces the path-dependent planning problem to an embedding problem, that is, constructing a McKean-Vlasov dynamics with given marginals (\mu 0,\mu 1). Some sufficient conditions on (\mu 0,\mu 1) are provided to guarantee the existence of solutions. We also characterize, up to integrability, the minimum entropy solution of the planning problem. In particular, as uniqueness does not hold anymore in our path-dependent setting, one can naturally introduce an optimal planning problem which would be reduced to an optimal transport problem along controlled McKean-Vlasov dynamics.

Original language | English (US) |
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Pages (from-to) | 1415-1437 |

Number of pages | 23 |

Journal | SIAM Journal on Control and Optimization |

Volume | 61 |

Issue number | 3 |

DOIs | |

State | Published - 2023 |

## Keywords

- McKean-Vlasov dynamic
- mean field games
- optimal transport
- planning problem

## ASJC Scopus subject areas

- Control and Optimization
- Applied Mathematics