A variational approach to the mean field planning problem

We investigate a first-order mean field planning problem of the form {−∂tu+H(x,Du)=f(x,m)in (0,T)×Rd,∂tm−∇⋅(mHp(x,Du))=0in (0,T)×Rd,m(0,⋅)=m0,m(T,⋅)=mTin Rd, associated to a convex Hamiltonian H with quadratic growth and a monotone interaction term f with polynomial growth. We exploit the variational structure of the system, which encodes the first order optimality condition of a convex dynamic optimal entropy-transport problem with respect to the unknown density m and of its dual, involving the maximization of an integral functional among all the subsolutions u of an Hamilton-Jacobi equation. Combining ideas from optimal transport, convex analysis and renormalized solutions to the continuity equation, we will prove existence and (at least partial) uniqueness of a weak solution (m,u). A crucial step of our approach relies on a careful analysis of distributional subsolutions to Hamilton-Jacobi equations of the form −∂tu+H(x,Du)≤α, under minimal summability conditions on α, and to a measure-theoretic description of the optimality via a suitable contact-defect measure. Finally, using the superposition principle, we are able to describe the solution to the system by means of a measure on the path space encoding the local behavior of the players.