On the stabilization of a vibrating equation (Q1961308)

The authors consider the following modelization of a flexible torque arm, controlled by the two feedback laws \(U_1\) and \(U_2\) to be determined such that the closed-loop system is asymptotically stable: \[ \begin{alignedat}{2} & y_{tt}(x,t)- (ay_x)_x(x,t)+\alpha y_t(x,t)+\beta y(x,t)= 0, &&\quad 0<x<1,\;t>0,\\ & (ay_x)(0,t)= \varepsilon_1 U_1(t), &&\quad t>0,\\ & (ay_x)(1,t)= \varepsilon_2 U_2(t), &&\quad t>0.\end{alignedat} \] It is shown that a nonlinear boundary velocity feedback is sufficient to stabilize asymptotically the system. Moreover, under growth conditions on the feedback, the uniform and the rational decay rate of the energy is also estimated.