Annals of Mathematical Sciences and Applications

Volume 8 (2023)

Number 3

Special Issue Dedicated to the Memory of Professor Roland Glowinski

Guest Editors: Annalisa Quaini, Xiaolong Qin, Xuecheng Tai, and Enrique Zuazua

Control theory on Wasserstein space: a new approach to optimality conditions

Pages: 565 – 628



Alain Bensoussan (International Center for Decision and Risk Analysis, Naveen Jindal School of Management, University of Texas, Dallas, Tx., U.S.A.; and School of Data Science, City University of Hong Kong)

Ziyu Huang (School of Mathematical Sciences, Fudan University, Shanghai, China)

Sheung Chi Phillip Yam (Department of Statistics, Chinese University of Hong Kong)


We study the deterministic control problem in the Wasserstein space, following the recent works of Bonnet and Frankowska, but with a new approach. One of the major advantages of our approach is that it reconciles the closed loop and the open loop approaches, without the technicalities of the traditional feedback control methodology. It allows also to embed the control problem in the Wasserstein space into a control problem in a Hilbert space, similar to the lifting method introduced by P. L. Lions, used already in our previous works. The Hilbert space is different from that proposed by P. L. Lions, and it allows to recover the control problem in the Wasserstein space as a particular case.


mean field type control problem, Wasserstein space, Hilbert space, Pontryagin maximum principle, forward-backward equations, value function, Bellman equation

2010 Mathematics Subject Classification

35R15, 49L25, 49N70, 60F99, 60H10, 60H15, 60H30, 91A13, 93E20

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This work is dedicated to the memory of Professor Roland Glowinski

Alain Bensoussan is supported by the National Science Foundation under grant NSF-DMS-2204795. This work also constitutes part of Ziyu Huang’s Ph.D. dissertation at Fudan University, China. Phillip Yam acknowledges the financial supports from HKGRF-14301321 with the project title “General Theory for Infinite Dimensional Stochastic Control: Mean Field and Some Classical Problems”, and HKGRF-14300123 with the project title “Well-posedness of Some Poisson-driven Mean Field Learning Models and their Applications.”

Received 29 May 2023

Accepted 21 September 2023

Published 14 November 2023