Uniform synchronization of an abstract linear second order evolution system

TL;DR

This paper investigates the conditions for uniform long-term synchronization of abstract linear second order evolution systems in Hilbert spaces, establishing minimal damping requirements and analyzing system structures.

Contribution

It provides a novel analysis of uniform synchronization over infinite horizons, including minimal damping bounds and algebraic conditions for system coupling.

Findings

Established lower bounds on damping for synchronization

Clarified algebraic structure and compatibility conditions

Applied results to wave and Kirchhoff plate systems

Abstract

Although the mathematical study on the synchronization of wave equations at finite horizon has been well developed, there was few results on the synchronization of wave equations for long-time horizon. The aim of the paper is to investigate the uniform synchronization at the infinite horizon for one abstract linear second order evolution system in a Hilbert space. First, using the classical compact perturbation theory on the uniform stability of semigroups of contractions, we will establish a lower bound on the number of damping, necessary for the uniform synchronization of the considered system. Then, under the minimum number of damping, we clarify the algebraic structure of the system as well as the necessity of the conditions of compatibility on the coupling matrices. We then establish the uniform synchronization by the compact perturbation method and then give the dynamics of the asymptotic orbit. Various applications are given for the system of wave equations with boundary feedback or (and) locally distributed feedback, and for the system of Kirchhoff plate with distributed feedback. Some open questions are raised at the end of the paper for future development. The study is based on the synchronization theory and the compact perturbation of semigroups.