International audienceWe consider minimal time control problems governed by an affine system w.r.t. the control. Our aim is to study properties of the optimalsynthesis in presence of a singular locus that involves a saturation point for the singular control. We show that the optimal synthesis exhibits a prior saturation point at the intersection of the singular arc and a switching curve,and we also discuss qualitative properties of this curve. We high light this phenomenon on several models arising in nuclear magnetic resonance andin bioprocesses