Control on Hilbert spaces and application to some mean field type control problems

We propose a new approach to studying classical solutions of the second order 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 lifting 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 (ESAIM Control Optim. Calc. Var. 25 (2019) 1–36).