Existence and robustness results of attractors for partially-damped piezoelectric beams

Abstract

This paper deals with a fully-dynamic piezoelectric beam model with a nonlinear force acting on the longitudinal displacements of the beam. The equations of motion follows a system of non-compactly coupled wave equation. As the magnetic effects are discarded by setting the magnetic permeability $ \mu $ to zero, the equations are fully decoupled to a single wave equation (electrostatic/quasi-static beam model). The existence of smooth finite-dimensional global attractors is proved by considering a damping injection to only one of the equations. Next, the existence of a finite set of determining functionals for the long-time behavior of the system is proved. Finally, to transition from the fully-dynamic beam model to the commonly-used electrostatic/quasi-static beam model, a singular limit problem $ \mu \to 0 $ is considered. This analysis allows to compare the attractors from one model to another in the sense of the upper semi-continuity of the attractor $ \mathcal{A}_{\mu} $ as $ \mu \to 0 $.