Modified Sobolev Spaces in Controllability Problems for the Wave Equation on a Half-Plane

The $2$-d wave equation $w_{tt}=\Delta w$, $t\in(0,T)$, on the half-plane $x_1>0$ controlled by the Neumann boundary condition $w_{x_1}(0,x_2,t)=\delta(x_2)u(t)$ is considered in Sobolev spaces, where $T>0$ is a constant and $u\in L^\infty(0,T)$ is a control. This control system is transformed into a control system for the $1$-d wave equation in modified Sobolev spaces introduced and studied in the paper, and they play the main role in the study. The necessary and sufficient conditions of (approximate) $L^\infty$-controllability are obtained for the $1$-d control problem. It is also proved that the $2$-d control system replicates the controllability properties of the $1$-d control system and vise versa. Finally, the necessary and sufficient conditions of (approximate) $L^\infty$-controllability are obtained for the $2$-d control problem.