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

TL;DR

This paper analyzes the stability of a piezoelectric beam under Coleman-Gurtin and Gurtin-Pipkin thermal laws, showing exponential decay without electrical control in Coleman-Gurtin case and polynomial stability with damping in Gurtin-Pipkin case.

Contribution

It provides a comprehensive stability analysis for piezoelectric beams with different thermal laws and damping mechanisms, highlighting conditions for exponential and polynomial decay.

Findings

Exponential decay achieved under Coleman-Gurtin law without electrical control.

Polynomial stability established with Gurtin-Pipkin law when electrical components are damped.

Different thermal laws lead to distinct stability behaviors in piezoelectric beams.

Abstract

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.