Near Optimality of Stochastic Control for Singularly Perturbed McKean-Vlasov Systems (Ji-Feng Zhang)

　　In this paper, we are concerned with the optimal control problems for a class of systems with fast-slow processes. The problem under consideration is to minimize a functional subject to a system described by a two-time scaled McKean--Vlasov stochastic differential equation whose coefficients depend on state components and their probability distributions. Firstly, we establish the existence and uniqueness of the invariant probability measure for the fast process. Then, by using the relaxed control representation and the martingale method, we prove the weak convergence of the slow process and control process in the original problem, and we obtain an associated limit problem in which the coefficients are determined by the average of those of the original problem with respect to the invariant probability measure. Finally, by establishing the nearly optimal control of the limit problem, we obtain the near optimality of the original problem.

　　Publication:

　　SIAM Journal on Control and Optimization, Vol. 60, Iss. 5 (2022), 10.1137/21M1466177

　　Author:

　　Yun Li

　　Key Laboratory of Systems and Control, Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

　　School of Mathematics Sciences, University of Chinese Academy of Sciences, Beijing 100149, China

　　Fuke Wu

　　School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei, 430074, P.R. China

　　Ji-Feng Zhang

　　Key Laboratory of Systems and Control, Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

　　School of Mathematics Sciences, University of Chinese Academy of Sciences, Beijing 100149, China

　　Email: jif@iss.ac.cn