A potential approach for mean-field planning problem

In this talk, I will discuss planning problems for first- and second-order one-dimensional mean-field games (MFGs). These games are comprised of a Hamilton–Jacobi equation coupled with a Fokker–Planck equation. Applying Poincare’s Lemma to the Fokker–Planck equation, we deduce the existence of a potential. Rewriting the Hamilton– Jacobi equation in terms of the potential, we obtain a system of Euler–Lagrange equations for certain variational problems. Instead of the mean-field planning problem (MFP), we study this variational problem. By the direct method in the calculus of variations, we prove the existence and uniqueness of solutions to the variational problem. We also consider a first-order MFP with congestion. We prove that the congestion problem has a weak solution by introducing a potential and relying on the theory of variational inequalities.