Nonlinear feedback control of Timoshenko beam
type: Item
instance of: scholarly article
title: Nonlinear feedback control of Timoshenko beam
zbMATH Open document ID: 0834.93023
published in: Science in China. Series A
publication date: 1996-04-01T00:00:00Z
review text: Recently, much attention was given to both the static and dynamic models of Timoshenko's beam. The authors consider boundary control of the dynamic model given by coupled equations: \[ \rho {\partial^2 w\over \partial t^2}- K {\partial^2 w\over \partial x^2}+ K {\partial\varphi\over \partial x}= 0,\quad t> 0,\quad 0\leq x< \ell, \] \[ I_p {\partial^2\varphi\over \partial t^2}- EI {\partial^2\varphi\over \partial x^2}+ K\Biggl(\varphi- {\partial w\over \partial x}\Biggr)= 0,\quad t> 0,\quad 0\leq x< \ell, \] \(w(0,t)= \varphi(0, t)= 0\), \[ K\varphi(\ell, t)- K {\partial w(\ell,t)\over \partial x} = f\Biggl({\partial w(\ell, t)\over \partial t}\Biggr),\;-EI {\partial\varphi(\ell,t)\over \partial x}= g\Biggl({\partial\varphi(\ell, t)\over \partial t}\Biggr), \] \(f\) and \(g\) are monotone increasing functions. For linear boundary control the problem of energy decay and controllability was investigated by Lagnese, Lasiecka, D. Russell and their collaborators. The authors assure nonlinear controls of the form: \(f= \alpha h\), \(g= \beta h\), where \(\alpha\), \(\beta\) are constants and \[ h(\xi)= \begin{cases} |\xi|^p \text{ sign } \xi\quad &\text{when }|\xi|\leq 1\\ \xi \quad &\text{when } |\xi|> 1.\end{cases} \] Thus, the boundary feedback is still linear for large values of the signals \({\partial w\over \partial t}\), \({\partial\varphi\over \partial t}\) at \(x= \ell\). The authors define an appropriate Sobolev space \(\mathcal H\) and rewrite the effects of the closed loop control system as a nonlinear evolution equation in \(\mathcal H\). \(\dot Y(t)= AY(t)\), \(Y(0)= Y_0\), where \(Y\) is the vector \(\{w(\cdot, t), \dot w(\cdot, t), \dot\varphi(\cdot, t)\}^T\). Existence is proved via a fixed point theorem, which in turn uses Yosida's approximation to guarantee the contractive mapping property. Uniqueness is derived from equicontinuity, which they show to be a consequence of Sobolev's imbedding theorem. Next, the authors introduce a commonly used and conveni…
reviewed by: Vadim Komkov
Mathematics Subject Classification ID: 93C25
Mathematics Subject Classification ID: 74K10
Mathematics Subject Classification ID: 93C10
zbMATH DE Number: 817809
zbMATH Keywords: stabilization
zbMATH Keywords: Timoshenko's beam
zbMATH Keywords: boundary control
zbMATH Keywords: nonlinear controls
zbMATH Keywords: fixed point theorem
author: Dexing Feng
author: Weitao Zhang
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 817809