Smoothness of solutions to the initial-boundary value problem for the telegraph equation on the half-line with a locally summable potential

Abstract: We study solutions to the system $u_{tt}-u_{xx}+q(x)u=0, x>0,t>0$; $u|{t=0}=u_t|{t=0}=0, x>0$; $u|_{x=0}=g(t), t>0$, with a locally summable Hermitian matrix-valued potential $q$ and a $C^{\infty}$-smooth $\mathbb C^n$-valued boundary control $g$ vanishing near the origin. We prove that the solution $u^{g}(\cdot,T)$ is a function from $W^2_1([0,T];\mathbb C^n)$ and that the control operator $W^T:g\mapsto u^g(\cdot,T)$ is an isomorphism in $L_2([0,T];\mathbb C^n)$, and, in the case that $q$ is from $L_2([0,T];\mathbb C^n)$, also an isomorphism in $H^2([0,T];\mathbb C^n)$.