CONTROL ON HILBERT SPACES AND APPLICATION TO SOME MEAN FIELD TYPE CONTROL PROBLEMS

• 成果类型: Article
• 署名作者: Bensoussan, Alain; Graber, P. Jameson; Yam, Sheung Chi Phillip
• 署名单位: University of Texas System; University of Texas Dallas; Baylor University; Chinese University of Hong Kong
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/24-AAP2060
• 发表日期: 2024
• 页码: 4085-4136
• 关键词: STOCHASTIC DIFFERENTIAL-EQUATIONS; games

摘要

We propose a new approach to studying classical solutions of the second order Bellman equation and master equation for mean field type control problems, using a novel form of the lifting idea introduced by P.-L. Lions. Rather than studying the usual system of Hamilton-Jacobi/Fokker-Planck PDEs using analytic techniques, we instead study a stochastic control problem on a specially constructed Hilbert space, which is reminiscent of a tangent space on the Wasserstein space in optimal transport. On this Hilbert space we can use classical control theory techniques, despite the fact that it is infinite-dimensional. A consequence of our construction is that the mean field type control problem appears as a special case. Thus we preserve the advantages of the lifting procedure, while removing some of the difficulties. Our approach extends previous work by two of the coauthors, which dealt with a deterministic control problem for which the Hilbert space could be generic ( ESAIM Control Optim. Calc. Var. 25 (2019) 1-36).