Optimal boundary control by displacement at one end of a string under a given elastic force at the other end

The problem of optimal boundary control by displacement at one end of a string in the presence of a specified force mode at the other end is studied in the sense of a generalized solution of the corresponding mixed initial- boundary value problem from a Sobolev space. The problem of choosing an optimal boundary control from the infinite number of admissible controls is solved. A generalized solution of the mixed initial-boundary value problem is constructed explicitly and the uniqueness of the solution is proved.