Polynomial stability of piezoelectric beams with magnetic effect and tip body

Abstract

AbstractIn 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–Phillips 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 Weyl'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. 347 (2010), 455–478).