Nonlinear Diffusion Governed by McKean--Vlasov Equation on Hilbert Space and Optimal Control

This paper deals with optimal feedback control of measure-valued solutions of nonlinear diffusion governed by McKean-Vlasov equations. Questions of existence, uniqueness, and regularity properties of measure-valued solutions are addressed. A class of feedback controls furnished with a weak topology is introduced, and some important topological properties of the attainable set corresponding to these controls are presented. We consider several typical control problems with objective functionals which are functions of measures and prove the existence of optimal controls.