Reachability and controllability problems for the heat equation on a half-axis

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.