# Reachability problems for a wave-wave system with a memory term

## TL;DR

This paper addresses the reachability problem for a coupled wave system with memory, providing explicit time estimates and applying the Hilbert Uniqueness Method, with implications for viscoelasticity.

## Contribution

It introduces a method to solve the reachability problem for a wave system with memory, including explicit time estimates using Ingham type results.

## Key findings

- Explicit time estimates depend on system parameters
- Reachability is achieved under certain restrictions
- Applicable to viscoelasticity models

## Abstract

We solve the reachability problem for a coupled wave-wave system with an integro-differential term. The control functions act on one side of the boundary. The estimates on the time is given in terms of the parameters of the problem and they are explicitly computed thanks to Ingham type results. Nevertheless some restrictions appear in our main results. The Hilbert Uniqueness Method is briefly recalled. Our findings can be applied to concrete examples in viscoelasticity theory.