Exponential decay of Timoshenko beam with locally distributed feedback (Q2745703)

The purpose of this important paper is to study a question concerning the exponential stabilization of a nonuniform Timoshenko beam with locally distributed controls. The beam equations are described as follows NEWLINE\[NEWLINE\rho\ddot w- (Kw')'+ (K\varphi)'+ u_1(x, t)= 0,\quad 0< x< \ell,\quad t> 0,NEWLINE\]NEWLINE NEWLINE\[NEWLINEI_\rho\ddot\varphi- (EI\varphi')'+ K(\varphi- w')+ u_2(x, t)= 0,NEWLINE\]NEWLINE NEWLINE\[NEWLINEw(0, t)= 0,\quad \varphi(0, t)= 0,NEWLINE\]NEWLINE NEWLINE\[NEWLINEK(\ell)[\varphi(\ell, t)- w'(\ell, t)]= 0,\quad EI\varphi'(\ell, t)= 0.NEWLINE\]NEWLINE The following locally distributed feedback controls are used to stabilize the vibration of the beam NEWLINE\[NEWLINEu_1(x, t)= \rho(x) b_1(x)\dot w(x, t),\quad u_2(x, t)= I_\rho b_2(x)\dot\varphi(x, t).NEWLINE\]NEWLINE Main result: The authors propose a precise proof of exponential stability of the closed-loop system. Without the assumption of different wave speeds, it is shown that, under some locally distributed controls, the vibration of the beam decays exponentially. The proof is obtained by using a frequency multiplier method.