  We study the optimal control of mean-field systems with heterogeneous and
asymmetric interactions. This leads to considering a family of controlled
Brownian diffusion processes with dynamics depending on the whole collection of
marginal probability laws. We prove the well-posedness of such systems and
define the control problem together with its related value function. We next
prove a law invariance property for the value function which allows us to work
on the set of collections of probability laws. We show that the value function
satisfies a dynamic programming principle (DPP) on the flow of collections of
probability measures. We also derive a chain rule for a class of regular
functions along the flows of collections of marginal laws of diffusion
processes. Combining the DPP and the chain rule, we prove that the value
function is a viscosity solution of a Bellman dynamic programming equation in a
$L^2$-set of Wasserstein space-valued functions.
