  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? Our viscous term
contains the fractional power of the Dirichlet Laplace operator and it is
multiplied by a small parameter devoted to tend to zero. 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 the vanishing parameter.
