The Problem of Two-Dimensional String Vibrations with a Nonlinear Condition

Abstract

We study a model of small spatial transverse vibrations of a string where the deviation of any of its points from the equilibrium is characterized by two coordinates. It is assumed that in the course of vibrations one end of the string is inside a bounded closed convex set С belonging to a plane @ perpendicular to the segment along which the string is stretched. In turn, the set C can move in the plane @, with its motion given by a mapping. The end of the string remains free until it touches the boundary of the set. After coming into contact, they move together. A formula representing the solution of the initial–boundary value problem describing this vibration process is obtained. The problem of boundary control of the vibration process is considered.