Contributions to Books:
M. Miletic, A. Arnold:
"An Euler-Bernoulli beam equation with boundary control: Stability and dissipative FEM";
in: "ASC Report 12/2013",
issued by: Institute for Analysis and Scientific Computing;
Vienna University of Technology,
We consider a model for the time evolution of a piezoelectric cantilever with tip mass. With appropriately shaped actuator and sensor electrodes, boundary control is applied and a passivity based feedback controller is used to include damping into the system. We assume that the cantilever can be modeled by the Euler-Bernoulli beam equation and obtain a coupled PDE-ODE system for the closed-loop control system. From the literature it is known that it is asymptotically stable. But in a refined analysis we shall
prove that this system is - somewhat surprisingly - not exponentially stable.
In the second part of this paper we derive a dissipative finite element method, based on piecewise cubic Hermitian shape functions and a Crank-Nicolson time discretization. We prove that this numerical method leads to energy dissipation, analogous to the continuous case. This is illustrated in a simulation example.
beam equation, boundary feedback control, asymptotic stability, dissipative Galerkin method, error estimates
Electronic version of the publication:
Created from the Publication Database of the Vienna University of Technology.