Control on Hilbert Space and Mean Field Control: the Common Noise Case

TL;DR

This paper extends the theory of control on Hilbert spaces to include common noise in mean field control problems, providing a complete framework for such stochastic control systems.

Contribution

It introduces an equivalent formulation of mean field control with common noise using Hilbert space methods, completing previous theories by including common noise effects.

Findings

Established a unique optimal control under new assumptions.

Proved the strict convexity and coercivity of the cost functional.

Extended the control theory to encompass common noise in mean field models.

Abstract

The objective of this paper is to provide an equivalent of the theory developed in P.~Cardaliaguet, F.~Delarue, J.M.~Lasry, P.L.~Lions \cite{CDLL}, following the approach of control on Hilbert spaces introduced by the authors in \cite{BGY-2}. We include the common noise in this paper, so the alternative is now complete. Since we consider a control problem, our theory applies only to Mean field control and not to mean field games. The assumptions are adapted to guarantee a unique optimal control, so they insure that the cost functional is strictly convex and coercive.