Asymptotic behaviour in the robot rendezvous problem

Abstract

<p>This paper presents a natural extension of the results obtained by Feintuch and Francis in (2012a,b) concerning the so-called robot rendezvous problem. In particular, we revisit a known necessary and sufficient condition for convergence of the solution in terms of Cesàro convergence of the translates S<sup>k</sup>x<sub>0</sub>, k≥0, of the sequence x<sub>0</sub> of initial positions under the right-shift operator S, thus shedding new light on questions left open in Feintuch and Francis (2012a,b). We then present a new proof showing that a certain stronger ergodic condition on x<sub>0</sub> ensures that the corresponding solution converges to its limit at the optimal rate O(t<sup>−1/2</sup>) as t→∞. After considering a natural two-sided variant of the robot rendezvous problem already studied in Feintuch and Francis (2012a) and in particular proving a new quantified result in this case, we conclude by relating the robot rendezvous problem to a more realistic model of vehicle platoons.</p>