二维Keller-Segel系统的平均场控制问题  

The meanfield control problem for the two-dimensional Keller-Segel system

作　　者：王昭 王宇澄 陈丽 Zhao Wang;Yucheng Wang;Li Chen

机构地区：[1]长春工业大学数学与统计学院,长春130012 [2]School of Business Informatics and Mathematics,University of Mannheim,Mannheim 68159,Germany

出　　处：《中国科学:数学》2025年第3期703-734,共32页Scientia Sinica:Mathematica

摘　　要：本文利用平均场极限理论得到带有随机效应的粒子相互作用系统的最优控制收敛于KellerSegel系统的最优控制,其中功效泛函具有相似的结构且包含平均场作用项.首先,为克服Poisson方程基本解的奇性,引入相对熵方法及依概率收敛得到粒子系统的解收敛于Keller-Segel系统的解;其次,结合容许控制函数空间的紧性得到功效泛函的平均场极限结果;最后,利用二维Keller-Segel系统的最优控制的存在性结合Γ-收敛技术观点,得到最优控制的平均场极限结果.In this paper,it is proved,by using the meanfield limit theory,that the optimal control of the twodimensional Keller-Segel system can be obtained as a limit of the corresponding control problem of the interacting stochastic particle system.In these problems,the cost functions are chosen in the same structure which includes the meanfield effect.The difficulty arises from the singularity of the fundamental solution of the two-dimensional Poisson equation.Therefore we start with the control problem of a smoothed version of the particle system.In the compactness argument,the Gamma-convergence technique has been used.The important technique step is to show that the compactness of the cost functions implies strong convergence of the meanfield limit.It is achieved by using a combination of the relative entropy method and the convergence in probability of the trajectory.

关 键 词：平均场控制 偏微分方程最优控制 相对熵方法 平均场极限 依概率收敛 

分 类 号：O231[理学—运筹学与控制论]