Asymptotic behaviour in the robot rendezvous problem

TL;DR

This paper reviews the asymptotic behavior of solutions in the robot rendezvous problem, providing new conditions for convergence and convergence rates, and relating it to vehicle platoon models.

Contribution

It offers a necessary and sufficient condition for convergence based on Cesàro means and establishes a convergence rate of O(t^{-1/2}) under stronger ergodic conditions.

Findings

Cesàro convergence characterizes solution convergence

Solutions can converge at a rate of O(t^{-1/2})

Connections to vehicle platoon models are discussed

Abstract

We present a non-technical overview of the results obtained by the authors (2015) concerning the so-called robot rendezvous problem studied by Feintuch and Francis (2012). In particular, we present a necessary and sufficient condition for convergence of the solution in terms of Ces\`aro convergence of the translates $S^k x_0$, $k\ge0$, of the sequence $x_0$ of initial positions under the right-shift operator $S$, thus shedding new light on questions left open in by Feintuch and Francis. We also formulate a stronger ergodic condition on $x_0$ which ensures that the corresponding solution converges to its limit at the rate $O(t^{-1/2})$ as $t\to\infty$. We conclude with a brief discussion of a natural two-sided variant of the robot rendezvous problem and by relating the robot rendezvous problem to a more realistic model of vehicle platoons, for which the authors' earlier general results lead to analogous statements.