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

Sergey Simonov

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

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 ; ; , with a locally summable Hermitian matrix-valued potential and a -smooth -valued boundary control vanishing near the origin. We prove that the solution is a function from and that the control operator is an isomorphism in , and, in the case that is from , also an isomorphism in .

Paper Structure

This paper contains 5 sections, 7 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. The minimal Schrödinger operator
  3. The initial-boundary value problem $\beta^T_{\rm v}$
  4. Smooth waves
  5. The control operator and the Sobolev norm

Key Result

Lemma 1

If $q\in \mathcal{L}_{1,\rm loc}([0,\infty);\mathbb M^n_{\mathbb C})$, then the integrals $\int_0^{\frac{t-x}{2}}q(\tau)q\left(\frac{t-x}{2}-\tau\right)d\tau$, $\int_x^{\frac{t+x}{2}}q(\tau)q\left(\frac{t+x}{2}-\tau\right)d\tau$ and $\int_0^xq(\tau)q\left(\frac{t-x}{2}+\tau\right)d\tau$ as functions

Theorems & Definitions (13)

  • Lemma 1
  • proof
  • Lemma 2
  • Remark 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Lemma 3
  • proof
  • ...and 3 more