Exponential stability results for the boundary-controlled fully-dynamic piezoelectric beams with various distributed and boundary delays

Abstract

We consider an infinite-dimensional model for the longitudinal vibrations of a fully-dynamic piezoelectric beam which strongly couples mechanical vibrations and fully dynamic electromagnetic effects due to Maxwell's equations. The corresponding model is known to be not exponentially stabilizable by only one boundary feedback controller, controlling the voltage at the electrodes or mechanical strains. We consider the same model with two boundary feedback controllers measuring tip velocity and total current accumulated at the electrodes. The main focus in this paper is to investigate the effects of (i) distributed or (ii) boundary-type delayed damping on the overall exponential stabilizability. First, the equations of motion in the state-space formulation are shown to be well-posed by the semigroup theory. Next, the Lyapunov theory is cleverly utilized to prove that the exponential stability of each model is retained if the coefficients of the delayed damping terms and the boundary feedback controllers satisfy explicit conditions. Concurrently, the electrostatic/quasi-static models for each type of delay are also investigated for comparison. Finally, simulations for low and high-frequency normal vibrational modes are provided to enforce the theoretical findings.