A variational approach to the mean field planning problem

We investigate a first-order mean field planning problem of the form {-partial derivative(t)u + H(x, Du) = f(x,m) in (0,T) x R-d , partial derivative(t)m - del.(mH(p) (x, Du)) = 0 in (0,T) x R-d , m(0,center dot) = m(0) , m(T, center dot) = m(T) in R-d, 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 -partial derivative(t)u + H(x, Du) <= alpha, under minimal summability conditions on alpha, 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.