Boundary stabilization of Timoshenko beam (Q2721955)

The author begins with the following and commonly accepted Timoshenko's model: NEWLINE\[NEWLINE\begin{aligned} & w_{tt}-(Kw')' +(K\varphi)'=0\;('\equiv \partial/\partial x),\;\ell>x>0,\;t>0,\\ & I_\rho \varphi_{tt}-EI \varphi''+K (\varphi-w')=0,\quad I_\rho= R\rho(x),\;EI(x)>0.\\ & \text{with }w(0,t)= \varphi(0,t)=0 \text{ (clamped at left end)}.\end{aligned}NEWLINE\]NEWLINE The boundary conditions at the right end, i.e. \(\{x\in (\ell,t)\}\), are NEWLINE\[NEWLINEK\varphi (\ell,t)-w' (\ell,t) \in f_1(\partial/ \partial t\;w(\ell,t)),\text{ and }-EI\varphi'(\ell,t)\in f_2(\partial/ \partial t w (\ell,t)).NEWLINE\]NEWLINEThe usual physical meanings assigned to the various symbols appearing here have been explained in previous reviews [see for example \textit{M. A. Shubov}'s article, ``Exact controllability of damped Timoshenko beam'', IMA J. Math. Control Inf. 17, 375-395 (2000; Zbl 0991.93016)]. Here \(f_1(\partial/ \partial t w (\ell,t))\), \(f_2(\partial/ \partial t \varphi (\ell,t))\) are the control force and control moment, respectively, applied at the free end. The only difference between the initial-boundary value problem posed by Shubov and by this author is Shubov's assumption of direct proportionality, rather than the choice of more general functions \(f_1\), \(f_2\) of the present author. These functions are assumed to be maximal monotone, and such that near zero the functions \(f_i(\xi)\) lie between two bounded exponential curves, neither one of which assumes value zero at zero or in the vicinity of zero.NEWLINENEWLINENEWLINEThe author defines the energy of the system \(E(t)\), which is in agreement with other papers on this topic, and then pursues a different and original path by designing a Lyapunov function \(F(t)=\mu tE(t) +G(t)\) such that \(G(t)\) depends on an integral of quadratic terms multiplied by constants, which in turn depend only on multiples of various quadratic terms. Use of Sobolev's embedding theorem and a lengthy and difficult calculation permit the author to conclude that \(F\) is indeed a Lyapunov function and to show that the total energy of the beam decreases exponentially under the proposed feedback control action.