Stabilization of piezoelectric beam with Coleman-Gurtin or Gurtin-Pipkin thermal law and under Lorenz gauge condition

In this paper, we present the analysis of stability for a piezoelectric beam subject to a thermal law (Coleman-Gurtin or Gurtin-Pipkin thermal law) adding some viscous damping mechanism to the electric field in $x-$direction and $z-$direction, and we discuss several cases. Then, there is no need to control the electrical field components in $x$-direction and $z-$ direction to establish an exponential decay of solutions when the beam is subjected to a Coleman-Gurtin law, otherwise a polynomial stability is established with Gurtin-Pipkin thermal law in case when the electrical field components are damped.