  In the paper, problems of controllability, approximate controllability,
reachability and approximate reachability are studied for the control system
$w_t=w_{xx}$, $w(0,\cdot)=u$, $x>0$, $t\in(0,T)$, where $u\in L^\infty(0,T)$ is
a control. It is proved that each end state of this system is approximately
reachable in a given time $T$, and each its initial state is approximately
controllable in a given time $T$. A necessary and sufficient condition for
reachability in a given time $T$ is obtained in terms of solvability a Markov
power moment problem. It is also shown that there is no initial state that is
null-controllable in a given time $T$. The results are illustrated by examples.
