  This paper studies the extended mean-field control problems with
state-control joint law dependence and Poissonian common noise. We develop the
stochastic maximum principle (SMP) and establish the connection to the
Hamiltonian-Jacobi-Bellman (HJB) equation on the Wasserstein space. The
presence of the conditional joint law in the McKean-Vlasov dynamics and its
discontinuity caused by the Poissonian common noise bring us new technical
challenges. To develop the SMP when the control domain is not necessarily
convex, we first consider a strong relaxed control formulation that allows us
to perform the first-order variation. We also propose the technique of
extension transformation to overcome the compatibility issues arising from the
joint law in the relaxed control formulation. By further establishing the
equivalence between the relaxed control formulation and the strict control
formulation, we obtain the SMP for the original problem in the strict control
formulation. In the part to investigate the HJB equation, we formulate an
auxiliary control problem subjecting to a controlled measure-valued dynamics
with Poisson jumps, which allows us to derive the HJB equation of the original
problem through an equivalence argument. We also show the connection between
the SMP and HJB equation and give an illustrative example of linear quadratic
extended mean-field control with Poissonian common noise.
