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.

在本文中，我们考虑存在聚集-扩散约束的平均场最优控制问题，在存在高斯噪声项的情况下，代理通过势相互作用。我们的分析集中在耦合汉密尔顿-雅各比方程和福克-普朗克方程的PDE系统上，分别描述了问题的最优控制方面和代理群体的演化。本文的主要贡献是上述系统解决方案的存在的结果。我们注意到该模型与均值博弈系统的理论紧密相关。但是，一个独特的特征涉及交互的非本地性。它影响福克-普朗克方程以及系统的哈密顿方程中的漂移项，从而导致需要解决的新困难。