This was part of Mathematical Advances in Mean Field Games

## Control on Hilbert Spaces and Application to Mean Field Control

Alain Bensoussan, University of Texas, Dallas
Thursday, December 16, 2021

Abstract: We discuss a new approach to mean field control theory, which replaces the control of a Fokker Planck equation by the control of a dynamic system, whose state evolves in a Hilbert space. However, this Hilbert space is not the space of square integrable random variables, introduced in the lifting method by P.L. Lions. The problem that we study can be seen as an extension of the standard mean field control problem, so we recover all the existing results of the standard theory, while preserving the advantages of working with functions on a Hilbert space, instead than functions on probability measures. The transformation obtained in the lifting method, is not exactly an extension. It is thus more complicated to recover the initial control problem, than in our approach.