Polynomial stability of piezoelectric beams with magnetic effect and tip body

TL;DR

This paper investigates the mathematical stability properties of a piezoelectric beam with magnetic effects and tip load, demonstrating polynomial decay of energy over time through advanced operator theory techniques.

Contribution

It introduces a novel analysis of the beam's stability, proving polynomial decay without exponential stability using a new abstract formulation.

Findings

The system is not exponentially stable.

The system exhibits polynomial stability.

Well-posedness is established via Lumer-Philips theorem.

Abstract

In this paper, we consider a dissipative system of one-dimensional piezoelectric beam with magnetic effect and a tip load at the free end of the beam, which is modeled as a special form of double boundary dissipation. Our main aim is to study the well-posedness and asymptotic behavior of this system. By introducing two functions defined on the right boundary, we first transform the original problem into a new abstract form, so as to show the well-posedness of the system by using Lumer-Philips theorem. We then divide the original system into a conservative system and an auxiliary system, and show that the auxiliary problem generates a compact operator. With the help of Wely's theorem, we obtain that the system is not exponentially stable. Moreover, we prove the polynomial stability of the system by using a result of Borichev and Tomilov (Math. Ann. {\bf 347} (2010), 455--478).