Tangency Property and Prior-Saturation Points in Minimal Time Problems in the Plane
ACTA APPLICANDAE MATHEMATICAE(2020)
Abstract
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.
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Key words
Geometric optimal control, Minimum time problems, Singular arcs
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