A Piezoelectric Euler-Bernoulli Beam with Dynamic Boundary Control: Stability and Dissipative FEM
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A Piezoelectric Euler-Bernoulli Beam with Dynamic Boundary Control: Stability and Dissipative FEM Maja Mileti´c · Anton Arnold
Received: 25 November 2013 / Accepted: 20 August 2014 © Springer Science+Business Media Dordrecht 2014
Abstract 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. Keywords Beam equation · Boundary feedback control · Asymptotic stability · Dissipative Galerkin method · Error estimates Mathematics Subject Classification (2010) 35B35 · 65M60 · 35P20 · 74S05 · 93D15 The authors were supported by the doctoral school PDE-Tech of TU Wien and the FWF-project I395-N16. The authors acknowledge a sponsorship by Clear Sky Ventures. We are grateful to A. Kugi and T. Meurer for introducing us to this topic, their help, and the many stimulating discussions.
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M. Mileti´c · A. Arnold ( ) Institute for Analysis and Scientific Computing, Technical University Vienna, Wiedner Hauptstr. 8-10, 1040 Vienna, Austria e-mail: [email protected] M. Mileti´c e-mail: [email protected]
M. Mileti´c, A. Arnold
1 Model The Euler-Bernoulli beam (EBB) equation with tip mass is a well-established model with a wide range of applications: for oscillations in telecommunication antennas, or satellites with flexible appendages [2, 5], flexible wings of micro air vehicles [8], and even vibrations of tall buildings due to external forces [31]. The interest of engineers and mathematicians in the corresponding control problems started in the 1980s. So various boundary control laws have been devised and mathematically analyzed in the literature—with
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