Mean Field Control for Diffusion Aggregation system with Coulomb InteractionThe mean field control problem for a multi-dimensional diffusion-aggregation system with Coulomb interaction is considered. The existence of optimal control is proved through the Gamma convergence of the control problem of a regularized particle control problem. The optimal control problem on the particle level is studied by using the corresponding Liouville equation. Because of strong aggregation effect, additional difficulties arises from control function in the well-posedness theory, so that the known method for multi-dimensional Keller-Segel equation can not be directly applied. We use a combination of local existence result and boot-strap argument to obtain the global solution with small initial data. Since we obtain a strong propagation of chaos result by combining the convergence in probability and relative entropy method, the compact support requirement for control functions, which has been often used in the literature, is not need.
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