# American Institute of Mathematical Sciences

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doi: 10.3934/eect.2020011

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

 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

Received  January 2019 Revised  May 2019 Published  August 2019

Fund Project: 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.

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.

Citation: Umberto Biccari, Mahamadi Warma. Null-controllability properties of a fractional wave equation with a memory term. Evolution Equations & Control Theory, doi: 10.3934/eect.2020011
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