This paper is devoted to the study of an optimal control problem under the presence of a saturation point on the singular locus. We consider the minimal time problem for a system describing a fed-batch reactor with one species and one substrate. Our aim is to find an optimal feedback control that steers the system to a given target in minimal time. The growth function is of Haldane type implying the existence of a singular arc. Unlike other studies on the minimal time problem governed by affine systems w.r.t. the control, we assume that the singular arc is non-necessary controllable everywhere. This brings interesting issues in terms of optimal synthesis. Thanks Pontryagin's Principle and numerical simulations, we provide an optimal synthesis of the problem.