Hierarchic control for the one-dimensional wave equation in domains with moving boundary

This paper addresses the study of the controllability for a one-dimensional wave equation in domains with moving boundary. This equation models the motion of a string where an endpoint is fixed and the other one is moving. When the speed of the moving endpoint is less than 1−21+e2, the controllability of this equation is established. We present the following results: the existence and uniqueness of Nash equilibrium, the approximate controllability with respect to the leader control, and the optimality system for the leader control.