Optimal control of McKean-Vlasov systems under partial observation and hidden Markov switching

We study a class of mean-field control problems under partial observation. The controlled dynamics are of McKean-Vlasov type and are subject to regime switching driven by a hidden Markov chain. The observation process depends on the control and on the joint distribution of the state and control, which prevents the direct application of standard filtering techniques. The main contribution of this paper is to show how this distribution dependence can be handled within a change-of-probability framework, leading to a well-posed separated control problem. We derive a Zakai equation with a specific structure for the unnormalized filter, and show that the corresponding value function satisfies a dynamic programming principle. This yields a Bellman equation posed on a convex subset of a Wasserstein space, characterizing the optimal control problem under partial observation.