Optimal boundary control by a force at one end of a string with a given force mode at the other end

We consider the problem of boundary control by a force applied to one end of a string in the case of a given force mode at the other end. The problem is studied in the sense of the generalized solution of the corresponding mixed initial-boundary value problem in the Sobolev space. We also solve the problem of choosing an optimal boundary control in the set of all admissible controls. The generalized solution of the mixed initial-boundary value problem is constructed in closed form, and its uniqueness is proved.