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 $W_1^2([0,T];{\mathbb{C}}^n)$ 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 $H^2([0,T];{\mathbb{C}}^n)$.