Null controllability of a system of viscoelasticity with a moving control

Chaves-Silva FW, Rosier L, Zuazua E (2014)

Publication Type: Journal article

Publication year: 2014

Journal

Journal De Mathematiques Pures Et Appliquees Elsevier

Book Volume: 101

Pages Range: 198-222

Journal Issue: 2

DOI: 10.1016/j.matpur.2013.05.009

Abstract

In this paper, we consider the wave equation with both viscous Kelvin-Voigt and frictional damping as a model of viscoelasticity in which we incorporate an internal control with a moving support. We prove the null controllability when the control region, driven by the flow of an ODE, covers all the domain. The proof is based upon the interpretation of the system as, roughly, the coupling of a heat equation with an ordinary differential equation (ODE). The presence of the ODE for which there is no propagation along the space variable makes the controllability of the system impossible when the control is confined into a subset in space that does not move. The null controllability of the system with a moving control is established in using the observability of the adjoint system and some Carleman estimates for a coupled system of a parabolic equation and an ODE with the same singular weight, adapted to the geometry of the moving support of the control. This extends to the multi-dimensional case the results by P. Martin et al. in the one-dimensional case, employing 1-d Fourier analysis techniques. © 2013 Elsevier Masson SAS.

Authors with CRIS profile

Enrique Zuazua Iriondo

Involved external institutions

Basque Center of Applied Mathematics (BCAM) ES Spain (ES) Institut national de Recherche en Informatique et en Automatique (INRIA) FR France (FR)

How to cite

APA:

Chaves-Silva, F.W., Rosier, L., & Zuazua, E. (2014). Null controllability of a system of viscoelasticity with a moving control. Journal De Mathematiques Pures Et Appliquees, 101(2), 198-222. https://dx.doi.org/10.1016/j.matpur.2013.05.009

MLA:

Chaves-Silva, Felipe W., Lionel Rosier, and Enrique Zuazua. "Null controllability of a system of viscoelasticity with a moving control." Journal De Mathematiques Pures Et Appliquees 101.2 (2014): 198-222.

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