Optimal Energy Control in Finite Time by varying the Length of the String

We consider a finite string where, at both end points, a homogeneous Dirichlet boundary condition holds. One boundary point is fixed, and the other is moving; hence the length of the string is changing in time. The string is controlled through the movement of this boundary point. We consider movements of the boundary that are Lipschitz continuous. Only movements for which at the given finite terminal time the string has the same length as at the beginning are admissible. Moreover, we impose an upper bound for the Lipschitz constant of the movement that is smaller than the speed of wave propagation. We consider the optimal control problem to find an admissible movement for which at the given terminal time the energy of the string is minimal. We give a sufficient condition for the existence and uniqueness of an optimal movement and construct an optimal control movement.