Asymptotic behaviour in the robot rendezvous problem

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 Cesàro convergence of the translates $S^k{x}_0$, $k\ge 0$, 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.

To appear in Automatica

References in corpus (2)

• Asymptotics for infinite systems of differential equations
• Asymptotic behaviour of coupled systems in discrete and continuous time

Cited by in corpus (1)

• Asymptotics and approximation of large systems of ordinary differential equations