Controllability Problems for the Heat Equation in a Half-Plane Controlled by the Neumann Boundary Condition with a Point-Wise Control

In the paper, the problems of controllability and approximate controllability are studied for the control system $w_t=\Delta w$, $w_{x_1}(0,x_2,t)=u(t)\delta(x_2)$, $x_1>0$, $x_2\in\mathbb R$, $t\in(0,T)$, where $u\in L^\infty(0,T)$ is a control. To this aid, it is investigated the set $\mathcal{R}_T(0)\subset L^2((0,+\infty)\times\mathbb R)$ of its end states which are reachable from $0$. It is established that a function $f\in\mathcal{R}_T(0)$ can be represented in the form $f(x)=g\big(|x|^2\big)$ a.e. in $(0,+\infty)\times\mathbb R$ where $g\in L^2(0,+\infty)$. In fact, we reduce the problem dealing with functions from $L^2((0,+\infty)\times\mathbb R)$ to a problem dealing with functions from $L^2(0,+\infty)$. Both a necessary and sufficient condition for controllability and a sufficient condition for approximate controllability in a given time $T$ under a control $u$ bounded by a given constant are obtained in terms of solvability of a Markov power moment problem. Using the Laguerre functions (forming an orthonormal basis of $L^2(0,+\infty)$), necessary and sufficient conditions for approximate controllability and numerical solutions to the approximate controllability problem are obtained. It is also shown that there is no initial state that is null-controllable in a given time $T$. The results are illustrated by an example.