On the Neumann boundary controllability for the non-homogeneous string on a segment

The control system $w_{tt}=w_{xx}-q(x)w$, $w_x(0,t)=u(t)$, $w_x(d,t)=0$, $x\in(0,d)$, $t\in(0,T)$, $d>0$, $0<T\leq d$ is considered. Here $q\in C^1[0,d]$, $q>0$, $q'_+(0)=q'_-(d)=0$, $u$ is a control, $|u(t)|\leq 1$ on $(0,T)$. The necessary and sufficient conditions of null-controllability and approximate null-controllability are obtained for this system. The controllability problems are considered in the modified Sobolev spaces. The controls that solve these problems are found explicitly. It is proved that among the solutions of the Markov trigonometric moment problem there are bang-bang controls solving the approximate null-controllability problem.