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We consider a stochastic control problem where the set of controls is not necessarily convex and the system is governed by a nonlinear backward stochastic differential equation. We establish necessary as well as sufficient conditions of optimality for two models. The first concerns the strict (classical) controls. The second is an extension of the first to relaxed controls, who are a measure valued processes.where b is given function, ξ is the terminal data and W = (W t ) t≥0 is a standard d-dimensional Brownian motion, defined on a filtered probability space Ω, F , (F t ) t≥0 , P satisfying the usual conditions. The control variable v = (v t ), called strict (classical) control, is an F t -adapted process with values in some set U of R k . We denote by U the class of all strict controls.The criteria to be minimized, over the set U, has the formLemma 13 (Ekeland's variational principle). Let (E, d) be a complete metric space and f : E −→ R be lower-semicontinuous and bounded from below. Given ε > 0, suppose u ε ∈ E satisfies f (u ε ) ≤ inf (f ) + ε. Then for any λ > 0, there exists v ∈ E such that

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We consider a control problem where the state variable is a solution of a stochastic differential equation (SDE) in which the control enters both the drift and the diffusion coefficient. We study the relaxed problem for which admissible controls are measure-valued processes and the state variable is governed by an SDE driven by an orthogonal martingale measure. Under some mild conditions on the coefficients and pathwise uniqueness, we prove that every diffusion process associated to a relaxed control is a strong limit of a sequence of diffusion processes associated to strict controls. As a consequence, we show that the strict and the relaxed control problems have the same value function and that an optimal relaxed control exists. Moreover we derive a maximum principle of the Pontriagin type, extending the well-known Peng stochastic maximum principle to the class of measure-valued controls.

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.(SDE for short) of the typewhere b, σ and G are given deterministic functions, x 0 is the initial data and W = (W t ) t≥0 is a d-dimensional standard Brownian motion, defined on a filtered probability space Ω, F , (F t ) t≥0 , P , satisfying the usual conditions. Theand η is an increasing process (componentwise), continuous on the left with limits on the right with η 0 = 0. The pair (q, η) is called mixed relaxed-singular control (relaxed control for short) and we denote by R the class of relaxed controls.The functional cost, to be minimized over R, has the form. Singular control problems have been studied by many authors includingBenȇs-Shepp-Witsenhausen [5] , Chow-Menaldi-Robin [8] , Karatzas-Shreve [18] , Davis-Norman [9] , Haussmann-Suo [14,15,16] . See [15] for a complete list of references on the subject. The approaches used in these papers, to solve the problem are mainly based on dynamic programming. It was shown in particular that the value function is solution of a variational inequality, and the optimal state is a reflected diffusion at the free boundary. Note that in [14] , the authors apply the compactification method to show existence of an optimal relaxed-singular control.The other major approach to solve control problems is to derive necessary conditions satisfied by some optimal control, known as the stochastic maximum principle. The first version of the stochastic maximum principle that covers singular control problems was obtained by , in which they consider linear dynamics, convex cost criterion and convex state constraints. necessary optimality conditions for non linear SDEs were obtained by Bahlali-Chala [1] and Bahlali-Mezerdi [2] .The common fact in this works is that an optimal strict singular control does not necessarily exist, the set U of strict singular controls (v, η), where v : [0, T ] × Ω −→ A 1 ⊂ R k , is too narrow and not being equipped with a good topological structure. The idea is then to introduce the class R of relaxed controls in which the controller chooses at time t, a probability measure q t (da) on the set A 1 , rather than an element v t of A 1 . The relaxed control problem find its interest in two essential points. The first is that it is ...