STABILITY OF EXACTLY SYNCHRONIZABLE STATE BY GROUPS FOR A COUPLED SYSTEM OF WAVE EQUATIONS WITH RESPECT TO APPLIED CONTROLS

For a coupled system of wave equations with locally distributed controls, we show that the state variable can be divided into three groups: the first group can exactly reach any given target, the second group can approximately reach any given target, while the last one evolutes following a conservative system, therefore, is independent of applied controls. In particular, the exactly synchronizable state by groups is independent of applied controls if and only if Kalman's minimum rank condition holds. This paper generalizes the previous work [13] from the simple case of exact synchronization to the general case of exact synchronization by groups.