Exponential decay rate for the energy of a flexible structure with dynamic delayed boundary conditions and a local interior damping

This note is dedicated to study the energy decay of a flexible structure with a local interior damping. The system consists of a platform and a flexible cable holding a load mass. Taking into account the dynamics of both ends of the cable and the presence of a delay term in the boundary, we manage to show that the presence of a localized distributed damping is strong enough to obtain the exponentially decay of the energy of the system. This improves the result obtained in Ammari and Chentouf (2020) where the damping was distributed over the entire spatial domain.