Asymptotic Behavior of Serially-Connected Piezoelectric-Elastic smart design: Enhanced Polynomial and new Exponential Decay

In [2], the authors explored the stability of longitudinal vibrations in transmission problems within the context of advanced system design—specifically, a serially connected piezoelectric-elastic configuration where local damping is applied exclusively to the elastic component. The nature of the solution's stability, whether polynomial or exponential, is governed entirely by the arithmetic properties of a key quotient involving all physical parameters. In this work, we refine and extend the polynomial energy decay rate established in [2,Theorem 1.3] by incorporating the results of Theorem 3.1. Furthermore, we investigate an alternative damping configuration and precisely determine the exponential decay rate. Numerical simulations corroborate our theoretical findings, offering insight into how the relative lengths of the piezoelectric and elastic components influence exponential stability.