Control in Hilbert Space and First Order Mean Field Type Problem

TL;DR

This paper extends control theory in Hilbert spaces to nonlinear, separable drift functions, establishing global solutions, optimality conditions, and applying these results to mean field control problems with Bellman equations.

Contribution

It provides the first global existence and uniqueness results for FBODEs with nonlinear, separable drift in Hilbert space and applies the lifting technique to mean field control problems.

Findings

Proved global existence and uniqueness of solutions to FBODEs in Hilbert space.

Established the sufficiency of the Pontryagin Maximum Principle.

Applied results to linear quadratic mean field control problems.

Abstract

We extend the work \cite{bensoussan2019control} by two of the coauthors, which dealt with a deterministic control problem for which the Hilbert space could be generic and investigated a novel form of the `lifting' technique proposed by P. L. Lions. In \cite{bensoussan2019control}, we only showed the local existence and uniqueness of solutions to the FBODEs in the Hilbert space which were associated to the control problems with drift function consisting of the control only. In this article, we establish the global existence and uniqueness of the solutions to the FBODEs in Hilbert space corresponding to control problems with separable drift function which is nonlinear in state and linear in control. We shall also prove the sufficiency of the Pontryagin Maximum Principle and derive the corresponding Bellman equation. Besides, we shall show an analogue in the stationary case. Finally, by using the `lifting' idea as in \cite{stochasticv2,stochasticv1}, we shall apply the result to solve the linear quadratic mean field type control problems, and to show the global existence of the corresponding Bellman equations.