U. Biccari, S. Micu Null-controllability properties of the wave equation with a second order memory termJ DIFFER EQUATIONS, Vol. 267, No. 2 (2019), pp. 1376-1422 doi.org/10.1016/j.jde.2019.02.009
Abstract: We study the internal controllability of a wave equation with memory in the principal part, defined on the one-dimensional torus \mathbb{T}=\mathbb{R}/2\pi\mathbb{Z} . We assume that the control is acting on an open subset \omega(t)\subset\mathbb{T} , which is moving with a constant velocity c\in\mathbb{R}\setminus\{-1,0,1\} . The main result of the paper shows that the equation is null controllable in a sufficiently large time T and for initial data belonging to suitable Sobolev spaces. Its proof follows from a careful analysis of the spectrum associated to our problem and from the application of the classical moment method.
