Аннотация:The longitudinal vibrations of a thin rectilinear elastic rod are studied. Based on the method of integro-differential relations developed by the authors, a generalized statement of the initial-boundary value problem is proposed, which solution is sought in terms of kinematic and dynamic variables defined in a Sobolev space. For a uniform rod controlled by external forces applied at its ends, an optimal control problem is considered to transfer the system to a terminal state in a fixed time. In this case, the minimized functional is a weighted sum of the mean energy stored by the rod during its motion and the square of a quadratic norm of control functions. By using D’Alembert’s representation, the solution to the related direct dynamic problem is derived in the form of traveling waves. By taking into account properties of the generalized solution, the control problem, which is considered in a two-dimensional space-time domain, is reduced to a one-dimensional quadratic variational problem. The latter has a finite set of traveling waves as unknowns. As a result, the optimal control law and the corresponding motion of the rod are obtained explicitly. Finally, integral characteristics of the motion are analyzed depending on the time horizon and the weight coefficient in the cost functional.