Boundary feedback stabilization of a novel bilinear and extensible piezoelectric beam model

Existing piezoelectric beam models in the literature mostly ignore the nonlinear contributions of longitudinal vibrations to the bending motions. However, these interactions substantially change the controllability and stabilizability at the high frequencies, and linear models fail to represent and predict the governing dynamics since mechanical nonlinearities are pronounced in certain applications. In this paper, a voltage-actuated nonlinear piezoelectric beam model, obtained by the electrostatic theory of Maxwell's equations, is considered. Unlike the existing literature, the model in the state-space formulation define a bilinear control system. By a carefully chosen PID-type stabilizing state feedback controllers, the solutions of the closed-loop system are proved to be exponentially stable by the combinations of multipliers technique and energy estimates. Unlike the fully elastic nonlinear beam model, the integral-type controller is a necessity.