Tangency Property and Prior-Saturation Points in Minimal Time Problems in the Plane
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Tangency Property and Prior-Saturation Points in Minimal Time Problems in the Plane T. Bayen1
· O. Cots2
Received: 19 September 2019 / Accepted: 18 June 2020 © Springer Nature B.V. 2020
Abstract In this paper, we consider minimal time problems governed by control-affinesystems 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. Keywords Geometric optimal control · Minimum time problems · Singular arcs
1 Introduction In this paper, we consider minimal time problems governed by single-input control-affinesystems in the plane x(t) ˙ = f (x(t)) + u(t) g(x(t)),
|u(t)| ≤ 1,
where f, g : R2 → R2 are smooth vector fields. Syntheses for such problems have been investigated a lot in the literature (see, e.g., [6, 14, 20, 25–27]). In particular, an exhaustive description of the various encountered singularities can be found in [14], as well as an algorithm leading to the determination of optimal paths. It is worth mentioning that even
B T. Bayen
[email protected] O. Cots [email protected]
1
Laboratoire de Mathématiques d’Avignon (EA 2151), Avignon Université, 84018 Avignon, France
2
Toulouse Univ., INP-ENSEEIHT, IRIT and CNRS, 2 rue Camichel, 31071 Toulouse, France
T. Bayen, O. Cots
though many techniques exist in this setting, the computation of an optimal feedback synthesis (global) remains in general difficult because of the occurrence of geometric loci such as singular arcs, switching curves, cut-loci... Our aim in this work is to focus on the notion of singular arc which appears in the synthesis when the switching function (the scalar product between the adjoint vector and the controlled vector field g) vanishes over a time interval. In the context of control-affine-systems, singular arcs can generically be explicitly retrieved by solving non-degenerate linear equations (see, e.g., [8]). Besides, in the two-dimensional case, the corresponding singular control us (which allows the associated trajectory to stay on the singular locus) can be expressed in feedback form x → us [x]. However, it may happen that us becomes non admissible, i.e., x → us [x] takes values above the maximal value for the control (namely 1 here). Such a situation naturally appears in several application models, see, e.g., [2, 3, 17, 22]. In that case, we say that a saturation phenomenon occurs
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