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

Paola Loreti; Daniela Sforza

doi:10.3934/eect.2020018

Article Contents

2020, Volume 9, Issue 1: 87-130. Doi: 10.3934/eect.2020018

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

Paola Loreti^, and
Daniela Sforza

Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza Università di Roma, Via Antonio Scarpa 16, 00161 Roma, Italy

^* Corresponding author: Paola Loreti
^* Corresponding author: Paola Loreti

Received: September 2018

Revised: June 2019

Early access: October 2019

Published: March 2020

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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.

Keywords:

Boundary observability,
reachability,
Fourier series,
hyperbolic integro-differential systems,
abstract linear evolution equations.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

Citation:

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Figure 1. $ P'(x) $ when $ P'(0)>0 $ and $ P'(x_0)<0 $

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Figure 2. $ P(x) $ when $ P(0)>0 $, $ P(x_1)>0 $ and $ P(x_2)>0 $

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