  In this paper, we consider stochastic optimal control of systems driven by
stochastic differential equations with irregular drift coefficient. We
establish a necessary and sufficient stochastic maximum principle. To achieve
this, we first derive an explicit representation of the first variation process
(in Sobolev sense ) of the controlled diffusion. Since the drift coefficient is
not smooth, the representation is given in terms of the local time of the state
process. Then we construct a sequence of optimal control problems with smooth
coefficients by an approximation argument. Finally, we use Ekeland's
variational principle to obtain an approximating adjoint process from which we
derive the maximum principle by passing to the limit.
