Null controllability of the structurally damped wave equation with moving point control

TL;DR

This paper proves that the one-dimensional structurally damped wave equation on a torus can be driven to zero using a moving point or interval control within a fixed time, regardless of initial conditions.

Contribution

It establishes null controllability for the damped wave equation with moving control, a novel result for this class of PDEs with moving control regions.

Findings

Null controllability holds in suitable Sobolev spaces.

Controllability is achieved in fixed positive time independent of initial data.

Results apply to control acting on a moving point or small interval.

Abstract

We investigate the internal controllability of the wave equation with structural damping on the one dimensional torus. We assume that the control is acting on a moving point or on a moving small interval with a constant velocity. We prove that the null controllability holds in some suitable Sobolev space and after a fixed positive time independent of the initial conditions.