Felix Klein Colloquium: Mean-Field Control for a Diffusion Aggregation System with Coulomb lnteraction

The mean-field control problem for a multi-dimensional diffusion-aggregation system with Coulomb interaction [the so called parabolic elliptic Keller-Segel system] is considered. The existence of optimal control is proved through the Gamma-convergence of the control problem of a regularized particle control problem. There are three building blocks in the whole argument.

Firstly, for the optimal control problem on the particle level, instead of using the classical method for a stochastic system, we study directly the control problem of a high-dimensional parabolic equation, i.e. the corresponding Liouville equation of the particle system.

Secondly, the strong propagation of chaos result for moderate interacting system is obtained by combining the convergence in probability and relative entropy method. Due to this strong mean field limit result, we avoid the compact support requirement for control functions, which has been often used in the literature.

Thirdly, because of strong the aggregation effect, additional difficulties arise from the control function in the well-posedness theory, so that the known method for the multi-dimensional Keller-Segel equation cannot be directly applied. lnstead, we use a combination of a local existence result and a bootstrap argument to obtain the global solution in the sub-critical regime.

Speaker: Prof. Dr. Li Chen, University of Mannheim, Germany

Time: 17:15 - 18:30 o'clock

Place: Building 48, room 210

The lectures of the Felix Klein Colloquium will be held at 17:15 in room 210 of the Mathematics Building 48. Beforehand - from 16:45 - there will be an opportunity to meet the speaker at the colloquium tea in room 580.