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

In this paper, we study the optimal control system driven by stochastic differential equations (SDEs) of mean-field type, in which the control variable has two components, the first being absolutely continuous and the second singular. On the other hand, the coefficients depend on the state of the solution process as well as of its expected value. Moreover, the cost functional is also of mean field type. This makes the control problem time inconsistent in the sense that the Bellman optimality principle does not hold. Our aim is to derive a stochastic maximum principle of optimal control of Pontriagin type to the class of measure-valued controls.