Null-controllability properties of the wave equation with a second order memory term

TL;DR

This paper investigates the null controllability of a wave equation with a memory term on a one-dimensional torus, demonstrating controllability with a moving control region under certain conditions.

Contribution

It establishes null controllability for a wave equation with memory on a torus with a moving control, extending previous results to include memory effects.

Findings

The wave equation with memory is null controllable in large enough time.

Controllability depends on the control region moving with a constant velocity.

Spectral analysis and moment method are key tools in the proof.

Abstract

We study the internal controllability of a wave equation with memory in the principal part, defined on the one-dimensional torus $\mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}$. We assume that the control is acting on an open subset $\omega(t)\subset\mathbb{T}$, which is moving with a constant velocity $c\in\mathbb{R}\setminus\{-1,0,1\}$. The main result of the paper shows that the equation is null controllable in a sufficiently large time $T$ and for initial data belonging to suitable Sobolev spaces. Its proof follows from a careful analysis of the spectrum associated to our problem and from the application of the classical moment method.