A piezoelectric Euler-Bernoulli beam with dynamic boundary control: stability and dissipative FEM

TL;DR

This paper analyzes the stability of a piezoelectric Euler-Bernoulli beam with dynamic boundary control, proving it is not exponentially stable, and develops a dissipative finite element method with proven convergence for numerical simulations.

Contribution

It provides a refined stability analysis showing the system is not exponentially stable and introduces a structure-preserving finite element method with error bounds.

Findings

The control system is asymptotically stable but not exponentially stable.

The finite element method dissipates energy similarly to the continuous system.

Error bounds and convergence rates are established and validated through simulations.

Abstract

We present a mathematical and numerical analysis on a control model for the time evolution of a multi-layered piezoelectric cantilever with tip mass and moment of inertia, as developed by Kugi and Thull [31]. This closed-loop control system consists of the inhomogeneous Euler-Bernoulli beam equation coupled to an ODE system that is designed to track both the position and angle of the tip mass for a given reference trajectory. This dynamic controller only employs first order spatial derivatives, in order to make the system technically realizable with piezoelectric sensors. From the literature it is known that it is asymptotically stable [31]. But in a refined analysis we first prove that this system is not exponentially stable. In the second part of this paper, we construct a dissipative finite element method, based on piecewise cubic Hermitian shape functions and a Crank-Nicolson time discretization. For both the spatial semi-discretization and the full x - t-discretization we prove that the numerical method is structure preserving, i.e. it dissipates energy, analogous to the continuous case. Finally, we derive error bounds for both cases and illustrate the predicted convergence rates in a simulation example.