Remarks on hierarchic control for the wave equation in moving domains

We study a Stackelberg strategy subject to the evolutionary linearized Kirchhoff equation for small vibrations of a stretched elastic string when the ends are variables. We assume that we can act in the dynamic of the system by a hierarchy of controls. According to the formulation given by H. von Stackelberg (see [3]), there are local controls, called followers, and global controls, called leaders. In fact, one considers situations where there are two cost (objective) functions. One possible way is to cut the control into two parts, one being thought of as “the leader” and the other one as “the follower”. This situation is studied in the paper, with one of the cost functions being of the controllability type. Existence and uniqueness is proven. The optimality system is given in the paper.