Asymptotics for Infinite Systems of Differential Equations

Abstract

Abstract. This paper investigates the asymptotic behaviour of solutions to certain infinite systems of ordinary differential equations. In particular, we use results from ergodic theory and the asymptotic theory of C0-semigroups to obtain a characterisation, in terms of convergence of certain Cesàro averages, of those initial values which lead to convergent solutions. Moreover, we obtain estimates on the rate of convergence for solutions whose initial values satisfy a stronger ergodic condition. These results rely on a detailed spectral analysis of the operator describing the system, which is made possible by certain structural assumptions on the operator. The resulting class of systems is sufficiently broad to cover a number of important applications, including in particular both the so-called robot rendezvous problem and an important class of platoon systems arising in control theory. Our method leads to new results in both cases.

Asymptotic Behaviour of Coupled Systems in Discrete and Continuous Time

“…In order to prepare the ground for the main result of this section, Theorem 2.7 below, we begin with a series of preliminary results, the first few of which are taken more or less directly from [8]. The first result gives a complete description of the spectrum of the operator T .…”

“…Following [8], we call the even integer n = n T appearing in this result the resolvent growth parameter. Here we let…”

“…We begin by observing that σ(T ) ⊂ D ∪ {1} and that, by the same argument as in [8,Section 2], the resolvent operator has the explicit form…”

“…The general approach taken in this paper can be viewed as a discrete counterpart to the authors' previous paper [8], in which the corresponding continuous-time problem is studied. As it turns out, the discrete setting presents its own challenges and requires new techniques but, in return, leads to optimal results which can even be used to improve the known results in the continuous setting.…”

“…In Section 3 we show explicitly how the general result can be applied both to this simple example and also to a more complex one in which the agents' state vectors consist not only of their positions but involve also a velocity component. In Section 4 we provide a link between the discrete and the continuous settings and show how Theorem 2.7 can be used to improve the main result of [8] in an important special case. Finally, in Section 5 we apply this improved result to give sharper rates of decay in the platoon model studied in [9,12,15].…”