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
