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

Li, Tatsien; Rao, Bopeng

ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 4, pp. 1675-1704.

The approximate boundary synchronization by $p$-groups $(p\ge 1)$ 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 $\left(N-p\right)$, 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.

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