caa a22 4500 606165967 CHVBK 20210128100656.0 cr unu---uuuuu 210128e20150801xx s 000 0 eng 10.1007/s10440-014-9965-1 doi (NATIONALLICENCE)springer-10.1007/s10440-014-9965-1 A Piezoelectric Euler-Bernoulli Beam with Dynamic Boundary Control: Stability and Dissipative FEM [Elektronische Daten] [Maja Miletić, Anton Arnold] 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 (Control and Observer Design for Nonlinear Finite and Infinite Dimensional Systems, Lecture Notes in Control and Information Sciences, pp.351-368, 2005). 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 (Kugi and Thull in Control and Observer Design for Nonlinear Finite and Infinite Dimensional Systems, Lecture Notes in Control and Information Sciences, pp.351-368, 2005). 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. Springer Science+Business Media Dordrecht, 2014 Beam equation nationallicence Boundary feedback control nationallicence Asymptotic stability nationallicence Dissipative Galerkin method nationallicence Error estimates nationallicence Miletić Maja Institute for Analysis and Scientific Computing, Technical University Vienna, Wiedner Hauptstr. 8-10, 1040, Vienna, Austria aut Arnold Anton Institute for Analysis and Scientific Computing, Technical University Vienna, Wiedner Hauptstr. 8-10, 1040, Vienna, Austria aut Acta Applicandae Mathematicae Springer Netherlands 138/1(2015-08-01), 241-277 0167-8019 138:1<241 2015 138 10440 https://doi.org/10.1007/s10440-014-9965-1 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10440-014-9965-1 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Miletić Maja Institute for Analysis and Scientific Computing, Technical University Vienna, Wiedner Hauptstr. 8-10, 1040, Vienna, Austria aut NATIONALLICENCE 700 1- Arnold Anton Institute for Analysis and Scientific Computing, Technical University Vienna, Wiedner Hauptstr. 8-10, 1040, Vienna, Austria aut NATIONALLICENCE 773 0- Acta Applicandae Mathematicae Springer Netherlands 138/1(2015-08-01), 241-277 0167-8019 138:1<241 2015 138 10440