An example for the switching delay feedback stabilization of an infinite dimensional system: The boundary stabilization of a string

For vibrating systems, a delay in the application of a feedback control may destroy the stabilizing effect of the control. In this paper we consider a vibrating string that is fixed at one end and stabilized with a boundary feedback with delay at the other end.We show that certain delays in the boundary feedback preserve the exponential stability of the system. In particular, we show that the system is exponentially stable with delays freely switching between the values 4L/c4L/c and 8L/c8L/c, where LL is the length of the string and cc is the wave speed.