Exponential stabilization of integral equations with singular kernels

The authors consider exponential stabilization of mechanical systems consisting of rigid and flexible members by feedback acting on the rigid parts. The flexible material is assumed to be linearly viscoelastic of convolution type with a completely monotone kernel. No better decay rate can be obtained by stabilization than the essential growth rate of the unperturbed system. A formula for the essential growth rate is given, and it is shown to depend only on the convolution kernel. It is negative (i.e. exponential stabilization is possible) if the kernel decays exponentially