Stabilization of Timoshenko beam with dissipative boundary feedback (Q2745591)

The authors study boundary control for the Timoshenko beam, as defined by the following equations NEWLINE\[NEWLINE\rho\partial w/\partial t+ K(\partial\varphi/\partial x- \partial^2w/\partial x^2)= 0,NEWLINE\]NEWLINE and NEWLINE\[NEWLINEI_\rho\partial\varphi/\partial t- EI\partial^2\varphi/\partial x^2+ K(\varphi-\partial w/\partial x)= 0,NEWLINE\]NEWLINE with zero conditions at the clamped end of the beam and with the following feedbacks applied at the free end \(x=\ell\): NEWLINE\[NEWLINE\begin{aligned} u_1(t) &= \alpha\partial w/\partial t(\ell,t)+ \beta\partial\varphi/\partial t(\ell, t),\\ u_2(t) &= \tau\partial w/\partial t(\ell, t)+ \gamma\partial\varphi/\partial t(\ell, t).\end{aligned}NEWLINE\]NEWLINE To the uninitiated it would appear that the same form of control has been duplicated at the same location. The authors fail to explain that the physical dimensions of \(\alpha\) and \(\tau\) and of \(\beta\) and \(\gamma\), respectively are different. Thus one control represents a force, while the other one a moment.NEWLINENEWLINENEWLINEThe authors prove that if the rank of the matrix \(B\) of these controls is 2 and \(B\) is positive, then they can exponentially stabilize the beam by a boundary feedback. This is an if and only if condition. They use the standard semigroup arguments, arguing that the linear evolution operator generates a \(C_0\) contraction semigroup. To simplify calculations they assume that \(\tau= \beta\) if the rank of \(B\) is 2. If the rank of \(B\) is 1, the authors derive necessary and sufficient conditions for the energy to decay to zero. The reviewer comments that \(\partial w/\partial x\) and \(\varphi\) have the same physical dimension (in fact in the static case \(\partial w/\partial x\) and \(\varphi\) have exactly the same meaning). Thus equating the coefficients of \(\partial w/\partial t\) and \(\partial\varphi/\partial t\) is not a trivial matter, which it appears to be in the authors' presentation.NEWLINENEWLINENEWLINESome of the calculations were rather ugly, and the authors quote names of colleagues who verified their validity.