Tangency property and prior-saturation points in minimal time problems in the plane

TL;DR

This paper analyzes minimal time control problems in the plane, revealing tangency properties at prior-saturation points and providing conditions for their occurrence, with applications to bioprocesses and MRI models.

Contribution

It introduces sufficient conditions for prior-saturation points and tangency properties in minimal time problems with singular controls, using Pontryagin's Maximum Principle.

Findings

Tangency of the bridge to the switching curve at prior-saturation points.

Non-linear equations for computing prior-saturation points.

Applications to bioprocess and MRI models.

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 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.