On a mean field optimal control problem

Abstract In this paper we consider a mean field optimal control problem with an aggregation–diffusion constraint, where agents interact through a potential, in the presence of a Gaussian noise term. Our analysis focuses on a PDE system coupling a Hamilton–Jacobi and a Fokker–Planck equation, describing the optimal control aspect of the problem and the evolution of the population of agents, respectively. The main contribution of the paper is a result on the existence of solutions for the aforementioned system. We notice this model is in close connection with the theory of mean-field games systems. However, a distinctive feature concerns the nonlocal character of the interaction; it affects the drift term in the Fokker–Planck equation as well as the Hamiltonian of the system, leading to new difficulties to be addressed.