Null-controllability properties of the wave equation with a second order memory term

Abstract We study the internal controllability of a wave equation with memory in the principal part, defined on the one-dimensional torus T = R / 2 π Z . We assume that the control is acting on an open subset ω ( t ) ⊂ T , which is moving with a constant velocity c ∈ R ∖ { − 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 with our problem and from the application of the classical moment method.