# American Institute of Mathematical Sciences

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
##### References:

show all references

##### References:
 [1] Bopeng Rao, Laila Toufayli, Ali Wehbe. Stability and controllability of a wave equation with dynamical boundary control. Mathematical Control & Related Fields, 2015, 5 (2) : 305-320. doi: 10.3934/mcrf.2015.5.305 [2] Mohamed Ouzahra. Controllability of the semilinear wave equation governed by a multiplicative control. Evolution Equations & Control Theory, 2019, 8 (4) : 669-686. doi: 10.3934/eect.2019039 [3] Ovidiu Cârjă, Alina Lazu. On the minimal time null controllability of the heat equation. Conference Publications, 2009, 2009 (Special) : 143-150. doi: 10.3934/proc.2009.2009.143 [4] Muhammad I. Mustafa. On the control of the wave equation by memory-type boundary condition. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1179-1192. doi: 10.3934/dcds.2015.35.1179 [5] Eugenio Sinestrari. Wave equation with memory. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 881-896. doi: 10.3934/dcds.1999.5.881 [6] Qi Lü, Enrique Zuazua. Robust null controllability for heat equations with unknown switching control mode. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4183-4210. doi: 10.3934/dcds.2014.34.4183 [7] Eduardo Cerpa. Null controllability and stabilization of the linear Kuramoto-Sivashinsky equation. Communications on Pure & Applied Analysis, 2010, 9 (1) : 91-102. doi: 10.3934/cpaa.2010.9.91 [8] Giacomo Dimarco. The moment guided Monte Carlo method for the Boltzmann equation. Kinetic & Related Models, 2013, 6 (2) : 291-315. doi: 10.3934/krm.2013.6.291 [9] Valentin Keyantuo, Mahamadi Warma. On the interior approximate controllability for fractional wave equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3719-3739. doi: 10.3934/dcds.2016.36.3719 [10] Meixia Dou. A direct method of moving planes for fractional Laplacian equations in the unit ball. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1797-1807. doi: 10.3934/cpaa.2016015 [11] Shirshendu Chowdhury, Debanjana Mitra, Michael Renardy. Null controllability of the incompressible Stokes equations in a 2-D channel using normal boundary control. Evolution Equations & Control Theory, 2018, 7 (3) : 447-463. doi: 10.3934/eect.2018022 [12] Farid Ammar Khodja, Cherif Bouzidi, Cédric Dupaix, Lahcen Maniar. Null controllability of retarded parabolic equations. Mathematical Control & Related Fields, 2014, 4 (1) : 1-15. doi: 10.3934/mcrf.2014.4.1 [13] Lianwen Wang. Approximate controllability and approximate null controllability of semilinear systems. Communications on Pure & Applied Analysis, 2006, 5 (4) : 953-962. doi: 10.3934/cpaa.2006.5.953 [14] Miaomiao Cai, Li Ma. Moving planes for nonlinear fractional Laplacian equation with negative powers. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4603-4615. doi: 10.3934/dcds.2018201 [15] Peng Gao. Null controllability with constraints on the state for the 1-D Kuramoto-Sivashinsky equation. Evolution Equations & Control Theory, 2015, 4 (3) : 281-296. doi: 10.3934/eect.2015.4.281 [16] Fulvia Confortola, Elisa Mastrogiacomo. Optimal control for stochastic heat equation with memory. Evolution Equations & Control Theory, 2014, 3 (1) : 35-58. doi: 10.3934/eect.2014.3.35 [17] Sho Matsumoto, Jonathan Novak. A moment method for invariant ensembles. Electronic Research Announcements, 2018, 25: 60-71. doi: 10.3934/era.2018.25.007 [18] Pierre Lissy. Construction of gevrey functions with compact support using the bray-mandelbrojt iterative process and applications to the moment method in control theory. Mathematical Control & Related Fields, 2017, 7 (1) : 21-40. doi: 10.3934/mcrf.2017002 [19] Piermarco Cannarsa, Genni Fragnelli, Dario Rocchetti. Null controllability of degenerate parabolic operators with drift. Networks & Heterogeneous Media, 2007, 2 (4) : 695-715. doi: 10.3934/nhm.2007.2.695 [20] El Mustapha Ait Ben Hassi, Farid Ammar khodja, Abdelkarim Hajjaj, Lahcen Maniar. Carleman Estimates and null controllability of coupled degenerate systems. Evolution Equations & Control Theory, 2013, 2 (3) : 441-459. doi: 10.3934/eect.2013.2.441

2018 Impact Factor: 1.048