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

In the paper, the equation of a vibrating non-homogeneous string, whose potential is not equal to a constant, is considered on a half-axis. The Neumann control of the class $L^\infty$ is considered at a point $x=0$. The control problem is studied in the Sobolev spaces. The sufficient conditions for nullcontrollability and approximate null-controllability at a free time $T>0$ are obtained for the given system. The controls solving these problems are found explicitly.