Control on Hilbert Spaces and Application to Mean Field Type Control Theory
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 HamiltonJacobi/FokkerPlanck 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.
 Publication:

arXiv eprints
 Pub Date:
 May 2020
 arXiv:
 arXiv:2005.10770
 Bibcode:
 2020arXiv200510770B
 Keywords:

 Mathematics  Probability;
 Mathematics  Analysis of PDEs;
 Mathematics  Optimization and Control