Stabilization results of a Lorenz piezoelectric beam with partial viscous dampings

TL;DR

This paper studies the stabilization of a Lorenz piezoelectric beam with partial viscous damping, proving stability under certain control conditions and validating results through numerical simulations.

Contribution

It introduces a new stabilization approach for Lorenz piezoelectric beams using partial viscous damping and control of the beam's center-line stretching.

Findings

Strong stability established under various conditions

Exponential stability achieved by controlling the beam's center-line stretch

Numerical results confirm theoretical stability analysis

Abstract

In this paper, we investigate the stabilization of a one-dimensional Lorenz piezoelectric (Stretching system) with partial viscous dampings. First, by using Lorenz gauge conditions, we reformulate our system to achieve the existence and uniqueness of the solution. Next, by using General criteria of Arendt-Batty, we prove the strong stability in different cases. Finally, we prove that it is sufficient to control the stretching of the center-line of the beam in x-direction to achieve the exponential stability. Numerical results are also presented to validate our theoretical result.