 
Summary: Proceedings in Applied Mathematics and Mechanics, 30 May 2011
EulerBernoulli Beam with Boundary Control: Stability and FEM
Maja Miletic1,
and Anton Arnold1,
1
Institut für Analysis und Scientific Computing, TU Wien, Wiedner Hauptstr. 810, A1040 Wien, Austria
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 designed to include damping
into the system. Assuming that the cantilever can be modeled by the EulerBernoulli beam equation, we obtain a coupled
PDEODE system. First we discuss its dissipativity, and its asymptotic but nonexponential stability. Next we derive a FEM
using piecewise cubic Hermitian shape functions that is still dissipative. This is illustrated on a numerical simulation.
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1 Model
We consider an EulerBernoulli beam on the interval [0, L], clamped at x = 0. The model reads (cf. [1]):
µutt + uxxxx = 0,
u(t, 0) = 0,
ux(t, 0) = 0,
Juxtt(t, L) + uxx(t, L) + U1 = 0,
Mutt(t, L)  uxxx(t, L) + U2 = 0,
with the feedback control law:
