# Null-controllability properties of a fractional wave equation with a memory term

## TL;DR

This paper investigates the null-controllability of a fractional wave equation with memory, demonstrating that with a moving control, the system can be driven to equilibrium within a large enough time frame.

## Contribution

It introduces a novel control strategy for a coupled nonlocal PDE-ODE system with memory, proving null controllability using spectral analysis and moment methods.

## Key findings

- System is null controllable in large enough time T
- Control acting on a moving subset achieves the goal
- Initial data in fractional Sobolev spaces can be driven to rest

## Abstract

We study the null-controllability properties of a one-dimensional wave equation with memory associated with the fractional Laplace operator. The goal is not only to drive the displacement and the velocity to rest at some time-instant but also to require the memory term to vanish at the same time, ensuring that the whole process reaches the equilibrium. The problem being equivalent to a coupled nonlocal PDE-ODE system, in which the ODE component has zero velocity of propagation, we are required to use a moving control strategy. Assuming that the control is acting on an open subset $\omega(t)$ which is moving with a constant velocity $c\in\mathbb{R}$, the main result of the paper states that the equation is null controllable in a sufficiently large time $T$ and for initial data belonging to suitable fractional order Sobolev spaces. The proof will use a careful analysis of the spectrum of the operator associated with the system and an application of a classical moment method.