Asymptotics for infinite systems of differential equations

TL;DR

This paper analyzes the long-term behavior of solutions to infinite systems of differential equations using ergodic theory and spectral analysis, providing convergence criteria, rate estimates, and applications in control theory.

Contribution

It introduces a spectral analysis approach combined with ergodic theory to characterize convergence and rates for solutions of infinite differential systems, with new results in control applications.

Findings

Characterization of initial values leading to convergent solutions

Estimates on the convergence rate under ergodic conditions

Application to robot rendezvous and platoon control systems

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 $C_0$-semigroups to obtain a characterisation, in terms of convergence of certain Ces\`aro 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.