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Null controllability of a system of viscoelasticity with a moving control
Authors:Felipe W. Chaves-Silva  Lionel Rosier  Enrique Zuazua
Affiliation:1. BCAM – Basque Center for Applied Mathematics, Mazarredo 14, 48009 Bilbao, Basque Country, Spain;2. Institut Elie Cartan, UMR 7502 UdL/CNRS/INRIA, B.P. 70239, 54506 Vandœuvre-lès-Nancy Cedex, France;3. Ikerbasque – Basque Foundation for Science, Alameda Urquijo 36-5, Plaza Bizkaia, 48011 Bilbao, Basque Country, Spain
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.
Keywords:Viscoelasticity   Decoupling   Moving control   Null controllability   Carleman estimates
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