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We propose a new approach to studying classical solutions of the Bellmanequation and Master equation for mean field type control problems, using anovel form of the "lifting" idea introduced by P.-L. Lions. Rather thanstudying the usual system of Hamilton-Jacobi/Fokker-Planck PDEs using analytictechniques, we instead study a stochastic control problem on a speciallyconstructed Hilbert space, which is reminiscent of a tangent space on theWasserstein space in optimal transport. On this Hilbert space we can useclassical control theory techniques, despite the fact that it is infinitedimensional. A consequence of our construction is that the mean field typecontrol problem appears as a special case. Thus we preserve the advantages ofthe lifiting procedure, while removing some of the difficulties. Our approachextends previous work by two of the coauthors, which dealt with a deterministiccontrol problem for which the Hilbert space could be generic.

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