Control on Hilbert Spaces and Application to Some Mean Field Type Control Problems

TL;DR

This paper introduces a novel Hilbert space framework for analyzing mean field type control problems, enabling classical control techniques to solve complex PDEs more effectively.

Contribution

It develops a new stochastic control approach on a specialized Hilbert space, simplifying the analysis of mean field control problems and extending previous deterministic methods.

Findings

The approach effectively solves Bellman and Master equations in mean field control.

It simplifies the analysis by avoiding traditional PDE techniques.

The method extends to a broader class of control problems.

Abstract

We propose a new approach to studying classical solutions of the Bellman equation and Master equation for mean field type control problems, using a novel form of the "lifting" idea introduced by P.-L. Lions. Rather than studying the usual system of Hamilton-Jacobi/Fokker-Planck PDEs using analytic techniques, we instead study a stochastic control problem on a specially constructed Hilbert space, which is reminiscent of a tangent space on the Wasserstein space in optimal transport. On this Hilbert space we can use classical control theory techniques, despite the fact that it is infinite dimensional. A consequence of our construction is that the mean field type control problem appears as a special case. Thus we preserve the advantages of the lifiting procedure, while removing some of the difficulties. Our approach extends previous work by two of the coauthors, which dealt with a deterministic control problem for which the Hilbert space could be generic.