Abstract
In this paper, we show both the necessity and the sufficiency of the compatibility condition on the coupling matrix for the exact synchronization by groups for a coupled system of wave equations with Dirichlet boundary controls. Moreover, if the coupling matrix is diagonalizable by blocks in a suitable basis, for example, as in the cases for a symmetric matrix, the state of synchronization by groups can be uniquely determined independently of the applied boundary controls. In the general case, the state of synchronization by groups depends on the boundary controls; however, we give an estimate of the difference between the state of synchronization by groups and the solution to a problem independent of boundary controls.
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