Energy decay rate of piezoelectric beams with damped electrical field in Lorenz gauge setting

In this study, we explore the stabilization of a one-dimensional piezoelectric stretching system incorporating partial viscous damping. Initially, we employ Lorenz gauge conditions to reformulate the system, ensuring the existence and uniqueness of the solution. We demonstrate polynomial stability with an energy decay rate of t-1/2 when a control is acting only on the electrical field component in the x-direction. However, exponential stability is established when damping is applied solely to the electrical field component in the z-direction. These results cover the open problems in (Akil et al. in Appl. Math. Optim. 87(2):26, 2023).