Asymptotics and approximation of large systems of ordinary differential equations

TL;DR

This paper investigates the asymptotic behavior of large infinite systems of coupled differential equations, providing conditions for boundedness, convergence rates, and implications for finite systems, with applications to the platoon problem.

Contribution

It introduces simplified conditions for boundedness and convergence in infinite systems, and offers a framework to analyze large finite systems with size-independent rates.

Findings

Established conditions for boundedness of the semigroup.

Derived sharp convergence rates to equilibrium.

Applied results to the platoon problem.

Abstract

In this paper we continue our earlier investigations into the asymptotic behaviour of infinite systems of coupled differential equations. Under the mild assumption that the so-called characteristic function of our system is completely monotonic we obtain a drastically simplified condition which ensures boundedness of the associates semigroup. If the characteristic function satisfies certain additional conditions we deduce sharp rates of convergence to equilibrium. We moreover address the important and delicate issue of the role of the infinite system in understanding the asymptotic behaviour of large but finite systems, and we provide a precise way of obtaining size-independent rates of convergence for families of finite-dimensional systems. Finally, we illustrate our abstract results in the setting of the well-known platoon problem.