On control of probability flows with incomplete information

The mean-field type control problems with incomplete information are considered. There are several points of view that can be adopted to study the dynamics in probability space. Eulerian framework describes probability flows by the continuity equation. Kantorovich formulation describes each probability flows in terms of a single distribution on the set of admissible trajectories. The superposition principle connects these frameworks for uncontrolled dynamics. In this article, a probability flow in the both frameworks must be generated by a control that based on incomplete information about state and/or the probability at every time instance. This article presents some links between these frameworks in the case of incomplete information. In particular, besides the convexity condition, the assumptions are founded that guarantees the equivalence between the Kantorovich and Eulerian framework. This expands [6, Theorem 1] to mean-field type control problem with incomplete information.