Mathematical Physics and Harmonic Analysis Seminar

Date: September 23, 2022

Time: 1:50PM - 2:50PM

Location: Bloc 306

Speaker: Alain Bensoussan, University of Texas at Dallas

Title: Control On Hilbert Spaces and Mean Field Control

Abstract: In this work, we describe an alternative approach to the general theory of Mean Field Control as presented in the book of P. Cardaliaguet, F. Delarue, J-M Lasry, P-L Lions: The Master Equation and the Convergence Problem in Mean Field Games, Annals of Mathematical Studies, Princeton University Press, 2019. Since it uses Control Theory and not P.D.E. techniques it applies only to Mean Field Control. The general difficulty of Mean Field Control is that the state of the dynamic system is a probability. Therefore, the natural functional space for the state is the Wasserstein metric space. P.L. Lions has suggested to use the correspondence between probability measures and random variables, so that the Wasserstein metric space is replaced with the Hilbert space of square integrable random variables. This idea is called the lifting approach. Unfortunately, this brilliant idea meets some difficulties, which prevents to use it as an alternative, except in particular cases. In using a different Hilbert space, we study a Control problem with state in a Hilbert space, which solves the original Mean Field Control problem, as a particular case, and thus provides a complete alternative to the approach of Cardaliaguet, Delarue, Lasry, Lions. Based on Joint work with P. J. GRABER, P. YAM. Research supported by NSF grants DMS- 1905449 and 2204795.