Generalized approximate boundary synchronization for a coupled system of wave equations

In this paper, we consider the generalized approximate boundary synchronization for a coupled system of wave equations with Dirichlet boundary controls. We analyse the relationship between the generalized approximate boundary synchronization and the generalized exact boundary synchronization, give a sufficient condition to realize the generalized approximate boundary synchronization and a necessary condition in terms of Kalman’s matrix, and show the meaning of the number of total controls. Besides, by the generalized synchronization decomposition, we define the generalized approximately synchronizable state, and obtain its properties and a sufficient condition for it to be independent of applied boundary controls.

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

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.