On the wave equation with the hysteresis type condition

In this paper we investigate the initial-boundary value problem describing the oscillation process with a hysteresis-type boundary condition. This kind of problem arises in modeling of the string oscillations, where the movement is restricted by a sleeve concentrated at one point $x=l.$ We suppose that the string is located along the segment $[0,l]$ and the sleeve can move in perpendicular to $ [0, l] $ direction. The analog of d'Alembert formula is obtained. A boundary control problem is analyzed for a small period of time. The boundary control problem is to find a control function allowing to put the oscillation process from the initial state to the given final state.