Nonlinear feedback control of Timoshenko beam
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 convenient sum of squared terms representing energy: \[ E(t)= \textstyle{{1\over 2}} |Y(t)|^2= \textstyle{{1\over 2}} \int^1_0 \{\rho |\dot w|^2+ I_p|\dot \varphi|^2+ EI|\varphi'|^2\}dx. \] The proof that \(\dot E\) is negative involves a rather delicate and lengthy calculation. Some energy estimates and stability of the closed loop system follow this computation.
scientific article; zbMATH DE number 817809