Asymptotic behaviour in the robot rendezvous problem

Abstract

Abstract. This paper presents a natural extension of the results obtained by Feintuch and Francis in [5,6] 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 k x0, k ≥ 0, of the sequence x0 of initial positions under the right-shift operator S, thus shedding new light on questions left open in [5,6]. We then present a new proof showing that a certain stronger ergodic condition on x0 ensures that the corresponding solution converges to its limit at the optimal rate O(t −1/2 ) as t → ∞. After considering a natural two-sided variant of the robot rendezvous problem already studied in [5] 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.

Asymptotic behavior of infinite systems determined by polynomial shift operators

“…In Feintuch and Francis [1], it is suggested that the Banach space 𝓁 ∞ should be chosen as the state space. The behavior of infinite chains of kinematic points and second-order infinite dimensional systems is considered in the 𝓁 ∞ -framework [1][2][3][4][5]. In Athalye et al [6], there appear infinite dimensional systems whose dynamics is governed by Laurent operators.…”

“…In Paunonen and Seifert [7], ergodic theory and C 0 ‐semigroups are used to investigate the asymptotic behavior of the infinite system. The resulting class of systems covers a number of important applications including the robot rendezvous problem [1,3,8] and platoon systems [9–14].…”

Weighted Consensus Problem for Multiagent Systems with Edge‐ and Node‐Weighted Directed Graphs

“…Synchronization phenomena were observed in di erent elds, such as synchronization of oscillators, swarms of birds, schools of sh, and robot rendezvous [1][2][3][4][5]. Synchronization is one of the fundamental aspects of selforganization in networks of systems.…”

“…e conventional, bipartite, and cluster consensus problems can be unified in our given weighted consensus frame. (2) To solve weighted consensus problem, the edge-and nodeweighted digraph is adopted to model the interaction topology among agents, by which the distributed protocol to solve the weighted consensus problem is provided. (3) e equivalence between weighted consensus and conventional consensus is established, by which most conventional consensus protocols can be generalized to solve the weighted consensus problem directly.…”