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

S. A. Simonov<sup>abc

a</sup> St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
^b Alferov Federal State Budgetary Institution of Higher Education and Science Saint Petersburg National Research Academic University of the Russian Academy of Sciences, St. Petersburg
^c St. Petersburg National Research University of Information Technologies, Mechanics and Optics

Abstract: We study solutions of the system
\begin{align*} &u_{tt}-u_{xx}+q(x)u=0, && x>0,\ t>0, &u|_{t=0}=u_t|_{t=0}=0, && x\ge0, &u|_{x=0}=f(t), && t\ge0, \end{align*}
with a locally summable Hermitian matrix-valued potential $q$ and a $\mathcal C^{\infty}$-smooth $\mathbb C^n$-valued boundary control $f$ vanishing near the origin. We show that the solution $u^f(\cdot,T)$ is a function from $\mathcal W^2_1([0,T];\mathbb C^n)$ and that the control operator $W^T:g\mapsto u^{g}(\cdot,T)$ is an isomorphism in $\mathcal L_2([0,T];\mathbb C^n)$, while for $q\in \mathcal L_2([0,T];\mathbb M^n_{\mathbb C})$ it is also an isomorphism in $\mathcal H^2([0,T];\mathbb C^n)$.

Key words and phrases: initial-boundary value problem, telegraph equation, matrix Schroedinger operator, BC-method, control operator, Goursat problem.

UDC: 517.958

Received: 05.10.2025