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

Umberto Biccari; Mahamadi Warma

doi:10.3934/eect.2020011

Article Contents

2020, Volume 9, Issue 2: 399-430. Doi: 10.3934/eect.2020011

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

Umberto Biccari^1, , and
Mahamadi Warma^2,

1.
DeustoTech, University of Deusto, 48007 Bilbao, Basque Country, Spain, Facultad de Ingeniería, Universidad de Deusto, Avenida de las Universidades 24, 48007 Bilbao, Basque Country, Spain
2.
University of Puerto Rico, Rio Piedras Campus, Department of Mathematics, Faculty of Natural Sciences, 17 University AVE. STE 1701 San Juan PR 00925-2537 (USA)

^* Corresponding author: Umberto Biccari
^* Corresponding author: Umberto Biccari

Received: January 2019

Revised: May 2019

Early access: August 2019

Published: June 2020

The work of the first author is supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement NO: 694126-DyCon), by the Grant MTM2017-92996-C2-1-R COSNET of MINECO (Spain), and by the ELKARTEK project KK-2018/00083 ROAD2DC of the Basque Government. The work of both authors is supported by the Air Force Office of Scientific Research (AFOSR) under Award NO: FA9550-18-1-0242

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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.

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Mathematics Subject Classification: 30E05, 35L05, 35R11, 45K05, 93B05, 93B60, 93C05.

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