Pointwise Damping and Time Delay in the Stabilization of a Mechanical System

It is considered a mechanical system composed of a string in cantilever having a pointwise damping. The discontinuous B(oundary) V(alue) P(roblem) is split in two coupled BVPs. Introducing the Riemann invariants and taking into account their invariance along the characteristics, a system of continuous time difference equations with two delays is associated to the coupled BVPs and a one-to-one correspondence between the solutions of the two mathematical objects is proven. Then a theorem on exponential stability for the difference system is obtained. The stability depends on the arithmetic properties of the delay ratio. The aforementioned one-to-one correspondence is then used to establish exponential stability of the classical solutions of the coupled BVPs. Some conclusions and a sketch of open and future research subjects end the paper. Copyright (c) 2024 The Authors.