Exponential stability of piezoelectric beams with delay and second sound

Abstract

In this paper, we consider a fully‐dynamic piezoelectric beam model subjected to a magnetic effect, where the heat flux is given by Cattaneo's law. It is well known that, in the absence of delay, the dissipation produced by the heat conduction is strong enough to make the piezoelectric beams exponentially stable. However, time delay effects may destroy this behavior. Here, we show the existence and uniqueness of solutions through the semigroup theory. Furthermore, under a smallness condition on the delay, we prove an exponential stability result via establishing the appropriate Lyapunov functional. Finally, we numerically illustrate the asymptotic behavior of the solution.