Infinite chains of kinematic points

Abstract

In formulating the stability problem for an infinite chain of cars, state space is traditionally taken to be the Hilbert space 2 , wherein the displacements of cars from their equilibria, or the velocities from their equilibria, are taken to be square summable. But this obliges the displacements or velocity perturbations of cars that are far down the chain to be vanishingly small and leads to anomalous behaviour. In this paper an alternative formulation is proposed wherein state space is the Banach space ∞ , allowing the displacements or velocity perturbations of cars from their equilibria to be merely bounded.

Asymptotics for Infinite Systems of Differential Equations

“…In particular, the Banach space we are working in is X = p (Z), where 1 ≤ p ≤ ∞. The following result, which can be viewed as an extension of the results in [9], is in large part a consequence of Theorem 4.3 but with slightly sharper estimates on the rates of decay. For 1 ≤ p ≤ ∞, we once again use the notation …”

“…As explained in the proof of [9,Theorem 3], for each t ≥ 0 there exists an integer n(t) ≥ 0 such that n(t) ≤ t ≤ n(t) + 1 and…”

“…In particular, C = 4.705 works for t ≥ 2, C = 2.191 works for t ≥ 100, and as t 0 → ∞ the value of the constant approaches e(2/π) 1/2 ≈ 2.169. (c) For further discussion of the robot rendezvous problem and in particular its connection with the theory of Borel summability, see [9,10].…”

“…We also show that for p = 1 and p = ∞ this statement is no longer true but our Theorem 5.1 provides a simple ergodic condition on the initial displacements of the vehicles which is necessary and sufficient for the solution to converge to a limit. Then, in Section 6 we return to the robot rendezvous problem and use our general result, Theorem 4.3, to settle several questions left open in [9,10]. We conclude in Section 7 by mentioning several topics which remain subjects for future research.…”

“…The first is the so-called robot rendezvous problem [9,10], where m = 1, A 0 = −1 and A 1 = 1. In this case the equations in (1.1) can be thought of as describing the motion in the complex plane of countably many vehicles, or robots, indexed by the integers k ∈ Z, following the rule that robot k moves in the direction of robot k −1 with speed equal to their separation.…”