  We consider in this paper, mixed relaxed-singular stochastic control
problems, where the control variable has two components, the first being
measure-valued and the second singular. The control domain is not necessarily
convex and the system is governed by a nonlinear stochastic differential
equation, in which the measure-valued part of the control enters both the drift
and the diffusion coefficients. We establish necessary optimality conditions,
of the Pontryagin maximum principle type, satisfied by an optimal
relaxed-singular control, which exist under general conditions on the
coefficients. The proof is based on the strict singular stochastic maximum
principle established by Bahlali-Mezerdi, Ekeland's variational principle and
some stability properties of the trajectories and adjoint processes with
respect to the control variable.
