Optimal boundary displacement control at one end of a string with a medium exerting resistance at the other end

We study a problem of optimal boundary control of vibrations of a one-dimensional elastic string, the objective being to bring the string from an arbitrary initial state into an arbitrary terminal state. The control is by the displacement at one end of the string, and a homogeneous boundary condition containing the time derivative is posed at the other end. We study the corresponding initial-boundary value problem in the sense of a generalized solution in the Sobolev space and prove existence and uniqueness theorems for the solution. An optimal boundary control in the sense of minimization of the boundary energy is constructed in closed analytic form.