Boundary displacement control for the oscillation process with boundary conditions of damping type for a time less than critical

In this paper, we study the problem of boundary control of string oscillations, which is carried out over a period of time less than the critical time. The control is performed by a displacement of one end of the string, whereas at the other end a uniform boundary condition with oblique derivative is given, and the direction of this derivative does not coincide with characteristics. The classical statement of the problem is considered. Necessary and sufficient conditions for the existence of a unique control are found and the control itself is obtained in an explicit analytical form.