On the approximate boundary synchronization for a coupled system of wave equations: direct and indirect controls^*

¹ School of Mathematical Sciences, Fudan University, Shanghai 200433, China ; Shanghai Key Laboratory for Contemporary Applied Mathematic; Nonlinear Mathematical Modeling and Methods Laboratory
² Département de Mathématique et d’Informatique, Institut de Recherche Mathématique Avancée, Université de Strasbourg, 67084 Strasbourg, France ; School of Mathematical Sciences, Fudan University, Shanghai 200433, China

^** Corresponding author: dqli@fudan.edu.cn

Received: 2 March 2016
Accepted: 7 June 2017

Abstract

The approximate boundary synchronization by p-groups (p ≥ 1) in the pinning sense has been introduced in [T.-T. Li and B. Rao, Asymp. Anal. 86 (2014) 199–224], in this paper the authors give a new and more natural definition on the approximate boundary synchronization by p-groups in the consensus sense for a coupled system of N wave equations with Dirichlet boundary controls. We show that the approximate boundary synchronization by p-groups in the consensus sense is equivalent to that in the pinning sense. Moreover, by means of a corresponding Kalman’s criterion, the concept of the number of total (direct and indirect) controls is introduced. It turns out that in the case that the minimal number of total controls is equal to (N − p), the existence of the approximately synchronizable state by p-groups as well as the necessity of the strong C_p-compatibility condition are the consequence of the approximate boundary synchronization by p-groups, while, in the opposite case, the approximate boundary synchronization by p-groups could imply some non-expected additional properties, called the induced approximate boundary synchronization.