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

Understand

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.