Infinitesimally helicoidal motions with fixed pitch of oriented geodesics of a space form

TL;DR

This paper investigates the controllability of a geometric control system involving infinitesimal helicoidal motions of oriented geodesics in three-dimensional space forms, establishing conditions for controllability and solving a related optimal switching problem.

Contribution

It characterizes controllability conditions for helicoidal motions in space forms and solves a minimal switching problem for connecting arbitrary geodesics.

Findings

Controllability holds if and only if alpha^2 ≠ kappa.

In spherical case with alpha = ±1, curves stay within fibers of a Hopf fibration.

Solved a minimal switching problem for connecting geodesics with admissible curves.

Abstract

Let L be the manifold of all (unparametrized) oriented lines of R^3. We study the controllability of the control system in L given by the condition that a curve in L describes at each instant, at the infinitesimal level, an helicoid with prescribed angular speed alpha. Actually, we pose the analogous more general problem by means of a control system on the manifold G_kappa of all the oriented complete geodesics of the three dimensional space form of curvature kappa: R^3 for kappa = 0, S^3 for kappa = 1 and hyperbolic 3-space for kappa = -1. We obtain that the system is controllable if and only if alpha ^2 not equal kappa. In the spherical case with alpha = (+/-) 1, an admissible curve remains in the set of fibers of a fixed Hopf fibration of S^3. We also address and solve a sort of Kendall's (aka Oxford) problem in this setting: Finding the minimum number of switches of piecewise continuous curves joininig two arbitrary oriented lines, with pieces in some distinguished families of admissible curves.