Mean-Field Control for Diffusion Aggregation Equation with Coulomb Interaction

.The mean-field control problem for a multidimensional diffusion–aggregation system with Coulomb interactions (the so-called parabolic elliptic Keller–Segel system) is considered. The existence of optimal control is proven through the \(Γ\) convergence of the corresponding control problem of the interacting particle system. There are three building blocks in the overall argument. First, for the optimal control problem at the particle level, instead of using the classical method for stochastic systems, we directly study the control problem of high-dimensional parabolic equations, specifically the Liouville equation. Second, we obtain strong propagation of chaos results for the interacting particle system by combining the convergence in probability and relative entropy methods. Owing to this strong mean-field limit result, we avoid imposing compact support requirements for control functions, which have often been used in the literature. Third, because of the strong aggregation effect, additional difficulties arise from the control function in obtaining the well-posedness theory of the diffusion–aggregation equation, making known methods inapplicable. Instead, we use a combination of local existence results and bootstrap arguments to obtain the global solution in the subcritical regime.Keywordsmean-field controloptimal control of PDErelative entropy methodmean-field limitconvergence in probabilityMSC codes35Q9360F1782C2260B10