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 occurrence 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.

在本文中，我们考虑了平面上由仿射系统控制的最小时间问题，并且我们将重点放在存在单个轨迹且存在单个控制饱和点的单个轨迹的综合问题上。在为数据提供足够的条件以确保出现先饱和点和切换曲线后，我们证明了电桥（即，则从奇异轨迹发出的最佳冲击弧在此点上与切换曲线在先饱和点相切。使用庞特里亚金最大原理证明了这一性质，该原理还提供了一组非线性方程式，可用于计算先验饱和点。这些问题在生物过程中的分批补料模型和磁共振成像（MRI）模型中得到了说明，针对这些模型讨论了点对点问题的最少时间合成。