Exponential stabilization of fully dynamic and electrostatic piezoelectric beams with delayed distributed damping feedback

Fully dynamic system of equations for a single piezoelectric beam strongly couples the mechanical (longitudinal) vibrations with the total charge distribution across the beam. Unlike the electrostatic (or quasi-static) assumption of Maxwell’s equations, the hyperbolic-type charge equations have been recently shown to affect the stabilizability of the high-frequency vibrational modes if one considers only a single boundary controller; voltage at the electrodes of the beam. In this paper, we consider viscously damped beam equations and a single distributed state feedback controller with a delay. The effect of the delay in the feedback is investigated for the overall exponential stabilizability dynamics of the piezoelectric beam equations. First, the equations of motion in the state-space formulation are shown to be well-posed by the semigroup theory. Next, an energy approach by the Lyapunov theory is utilized to prove that the exponential stability is retained only if the coefficient of the delayed feedback is strictly less than the coefficient of the state feedback. Finally, the results are compared to the ones of the electrostatic case.