Exponential stabilization of a smart piezoelectric composite beam with only one boundary controller

Layered smart composite beams involving a piezoelectric layer are traditionally actuated by a voltage source by the extension mechanism. In this paper, we consider only the bending and shear of a cantilevered piezoelectric smart composite beam modeled by the Mead-Marcus sandwich beam assumptions. Uniform exponential stabilitization with only one boundary state feedback controller, simultaneously controlling both bending moment and shear, is proved by using a spectral multiplier approach. The state feedback controller slightly differs from the classical counterparts by a non-trivial compact and nonnegative integral operator. This is due to the strong coupling of the charge equation with the stretching and bending equations. For simulations, the so-called filtered semi-discrete finite difference scheme is adopted.