Asymptotic behaviour of the solutions and optimal location of the actuator for the pointwise stabilization of a string (Q2774104)

Consider the mixed initial boundary value problem NEWLINE\[NEWLINEu_{tt}-u_{xx}+ u_t\delta_\xi =0,\;0<x<\pi,\;t>0,NEWLINE\]NEWLINE NEWLINE\[NEWLINEu(0,t)= u_x(\pi,t)=0,\;t>0,NEWLINE\]NEWLINE NEWLINE\[NEWLINE u(x,0)= u^0(x),\;u_t(x,0)= u^1(x),\;0<x<\pi,NEWLINE\]NEWLINE where \(\delta_\xi\) is the Dirac mass concentrated in the point \(\xi\in (0,\pi)\). It is proved that the problem is well posed in \(V\times L^2(0,\pi)\) where \(V=\{\varphi\in H^1(0,\pi)\), \(\varphi (0)=0\}\). Another result is that all solutions of finite energy have exponential decay provided \(\xi=(p/q)\pi\) with \(p\) and \(q\) coprime and \(p\) odd. An estimate of the decay speed is given showing that the optimum choice of \(p/q\) is \(1/2\).