  In this paper, we first consider the well-posedness and asymptotic behavior
of a one-dimensional piezoelectric beam system with control boundary conditions
of fractional derivative type, which represent magnetic effects on the system.
By introducing two new equations to deal with control boundary conditions of
fractional derivative type, we obtain a new equivalent system, so as to show
the well-posedness of the system by using Lumer-Philips theorem. We then prove
the lack of exponential stability by a spectral analysis, and obtain the
polynomial stability of the system by using a result of Borichev and Tomilov
(Math. Ann. {\bf 347} (2010), 455--478). To find a more stable system, we then
consider the stability of the above system with additional thermal effects
described by Fourier's law, and achieve the exponential stability for the new
model by using the perturbed functional method.
