Cosine operator and controllability of the wave equation with memory revisited

(Γ is relatively open in ∂Ω. The case Γ = ∂Ω is not excluded). The function f is a control which we use to steer the initial datum w0 ∈ L(Ω) to a target ξ ∈ L(Ω) at a certain time T . This kind of control problem has been studied by several authors and with different methods, since a system of the form (1) is important for the applications in viscoelasticity, thermodynamics of materials with memory and nonfickian diffusion. Note that in viscoelasticity also controllability of the pair (w,w) of the deformation and velocity has to be studied but here for simplicity we confine ourselves to the controllability of the sole component w. It is a fact that: 1) the controllability of the sole component w is sufficient for the solutions of source identification problems, see [11, 9, 12]; 2) also the controllability of the pair of the deformation and the stress (or the flux) has its interest, and this is a new problem which appears in the case of systems with memory, see [1, 2, 13]. The key idea which underlines essentially all the papers on controllability of Eq. (1) is that the controllability properties of the associated wave equation