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

TL;DR

This paper demonstrates the exponential stabilization of a cantilevered piezoelectric composite beam using a single boundary controller, effectively managing bending and shear despite complex coupling effects.

Contribution

It introduces a novel boundary feedback control method that stabilizes the beam with only one controller, accounting for strong coupling in the system.

Findings

Proves uniform exponential stabilization with one boundary controller

Designs a boundary feedback controller involving a non-trivial integral operator

Uses a filtered semi-discrete finite difference scheme for simulations

Abstract

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.