  In this paper, we consider minimal time problems governed by
control-affine-systems in the plane, and we focus on the synthesis problem in
presence of a singular locus that involves a saturation point for the singular
control. After giving sufficient conditions on the data ensuring occurence of a
prior-saturation point and a switching curve, we show that the bridge (i.e.,
the optimal bang arc issued from the singular locus at this point) is tangent
to the switching curve at the prior-saturation point. This property is proved
using the Pontryagin Maximum Principle that also provides a set of non-linear
equations that can be used to compute the prior-saturation point. These issues
are illustrated on a fed-batch model in bioprocesses and on a Magnetic
Resonance Imaging (MRI) model for which minimal time syntheses for the
point-to-point problem are discussed.
