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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, Wien, 2013, ISBN: 978-3-902627-06-3, 1 - 33.

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

http://www.asc.tuwien.ac.at/preprint/2013/asc12x2013.pdf