  We consider optimal control problems for systems governed by mean-field
stochastic differential equations, where the control enters both the drift and
the diffusion coefficient. We study the relaxed model, in which admissible
controls are measure-valued processes and the relaxed state process is driven
by an orthogonal martingale measure, whose covariance measure is the relaxed
control. This is a natural extension of the original strict control problem,
for which we prove the existence of an optimal control. Then, we derive
optimality necessary conditions for this problem, in terms of two adjoint
processes extending the known results to the case of relaxed controls.
