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In this paper, we study the optimal control system driven by stochasticdifferential equations (SDEs) of mean-field type, in which the control variablehas two components, the first being absolutely continuous and the secondsingular. On the other hand, the coefficients depend on the state of thesolution process as well as of its expected value. Moreover, the costfunctional is also of mean field type. This makes the control problem timeinconsistent in the sense that the Bellman optimality principle does not hold.Our aim is to derive a stochastic maximum principle of optimal control ofPontriagin type to the class of measure-valued controls.

The Relaxed Stochastic Maximum Principle in the Mean-field Singular Controls