Abstract
In this paper, we investigate the structure of reachable sets from a given point q 0 for a class of analytic control affine systems characterized, among other things, by possessing two singular trajectories initiating at q 0. The aim of the paper is to establish the connection between the minimal number of analytic functions needed for describing reachable sets and the number of geometrically optimal singular trajectories. The paper is written in a language of the sub-Lorentzian geometry. Also, the sub-Lorentzian geometry methods are used to prove theorems.
Highlights
This paper is a continuation of the research started in [8, 9] and devoted to the study on reachable sets for noncontact sub-Lorentzian structures on R3, as well as for affine control systems induced by them
As in [8, 9] our objective is to investigate the interrelation of the structure of reachable sets from a given point q0 for the mentioned systems and geometric optimality of singular trajectories starting at q0 or—speaking in the subLorentzian language—geometric optimality of timelike abnormal curves starting at q0
Sniadeckich 8, 00-950 Warszawa, Poland trajectory of a control system starting from a point q0 is said to be geometrically optimal if it is contained in the boundary of the reachable set from q0; cf. [1])
Summary
This paper is a continuation of the research started in [8, 9] and devoted to the study on reachable sets for noncontact sub-Lorentzian structures on R3, as well as for affine control systems induced by them. Trajectory of a control system starting from a point q0 is said to be geometrically optimal if it is contained in the boundary of the reachable set from q0; cf [1]). Note that unlike the Lorentzian case, the boundary ∂ ̃J +(q0, U ) (here and below, ∂ ̃ means the boundary with respect to U ) may contain timelike curves starting from q0. It can be proved [6] that such curves are abnormal curves for the underlaying distribution H (see [11] for a definition); they are Goh curves (cf [1]), but we do not need this latter fact in this paper. One makes sure that if ∇H φ is null f.d. on U and γ : [a, b] −→ U is t.f.d. (nspc.f.d.), the function [a, b] t −→ φ(γ (t)) is decreasing (nonincreasing)
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