On a non-linear wave equation and the control of an elastic string from one equilibrium location to another

This paper concerns a non-linear system of wave equations describing the motion in space of an elastic string. We derive the equations, determine the equilibrium solutions and, using the methods of quasilinear hyperbolic systems, prove that the natural initial, boundary value problem has classical solutions existing in neighbourhoods of the “stretched” equilibrium solutions. We then prove that the positions of the endpoints of the string can be controlled in such a way that the string moves from an equilibrium in one location to an equilibrium in another location.