Approximate Controllability Problems for the Heat Equation in a Half-Plane Controlled by the Dirichlet Boundary Condition with a Bounded Control

In the paper, the problems of approximate controllability are studied for the control system $w_t=\mathrm{\Delta }w$, $w(0,x_2,t)=u(x_2,t)$, $x_1\in {\mathbb{R}}_{+}=(0,+\mathrm{\infty })$, $x_2\in \mathbb{R}$, $t\in (0,T)$, where $u$ is a control belonging to a special subset of $L^{\mathrm{\infty }}(\mathbb{R}\times (0,T))\cap L^2(\mathbb{R}\times (0,T))$. It is proved that each initial state belonging to $L^2({\mathbb{R}}_{+}\times \mathbb{R})$ is approximately controllable to an arbitrary end state belonging to $L^2({\mathbb{R}}_{+}\times \mathbb{R})$ by applying these controls. A numerical algorithm of solving the approximate controllability problem for this system is given. The results are illustrated by an example.