Volume 24, Number 4, October–December 2018
|Page(s)||1675 - 1704|
|Published online||25 January 2019|
On the approximate boundary synchronization for a coupled system of wave equations: direct and indirect controls*
School of Mathematical Sciences, Fudan University,
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: firstname.lastname@example.org
Accepted: 7 June 2017
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 Cp-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.
Mathematics Subject Classification: 93B05 / 93B07 / 93C20 / 35L53
Key words: Direct and indirect controls / system of wave equations / Dirichlet boundary controls / approximate boundary synchronization by groups / strong Cp-compatibility condition / Kalman’s criterioninduced approximate synchronization
© EDP Sciences, SMAI 2018
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