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Asymptotics for infinite systems of differential equations

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Details

Original languageEnglish
Pages (from-to)1153-1178
Number of pages26
JournalSIAM Journal on Control and Optimization
Volume55
Issue number2
DOIs
Publication statusPublished - 2017
Publication typeA1 Journal article-refereed

Abstract

This paper investigates the asymptotic behavior 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 characterization, 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.

Keywords

  • Asymptotic behavior, C0-semigroup, Ergodic theory, Ordinary differential equations, Rates of convergence, Spectrum, System

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Field of science, Statistics Finland

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