Transformation operators and modified Sobolev spaces in controllability problems on a half-axis

In the paper, the control system $ w_{tt}=\frac1\rho(k w_x){}_x+\gamma w$, $w_x(0,t)=u(t)$, $x>0$, $t\in(0,T)$, is considered in special modified spaces of Sobolev type. Here $\rho$, $k$, and $\gamma$ are given functions on $[0,+\infty)$; $u\in L^\infty(0,\infty)$ is a control; $T>0$ is a constant. The growth of distributions from these spaces depends on the growth of $\rho$ and $k$. With the aid of some transformation operators, it is proved that the control system replicates the controllability properties of the auxiliary system $ z_{tt}=z_{\xi\xi}-q^2z$, $z_\xi(0,t)=v(t)$, $\xi>0$, $t\in(0,T)$, and vise versa. Here $q\ge0$ is a constant and $v\in L^\infty(0,\infty)$ is a control. For the main system, necessary and sufficient conditions of the $L^\infty$-controllability and the approximate $L^\infty$-controllability are obtained from those known for the auxiliary system.