The aim of this paper is to answer the question: Do the controls of a vanishing viscosity
approximation of the one dimensional linear wave equation converge to a control of the
conservative limit equation? The characteristic of our viscous term is that it contains
the fractional power α of the Dirichlet Laplace operator. Through the
parameter α we may increase or decrease the strength of the high
frequencies damping which allows us to cover a large class of dissipative mechanisms. The
viscous term, being multiplied by a small parameter ε devoted to tend to
zero, vanishes in the limit. Our analysis, based on moment problems and biorthogonal
sequences, enables us to evaluate the magnitude of the controls needed for each eigenmode
and to show their uniform boundedness with respect to ε, under the
assumption that α∈[0,1)\{½}. It follows that, under this assumption, our starting
question has a positive answer.