In this paper, an existence and uniqueness result of a class of second-order sweeping processes, with velocity in the moving set under perturbation in infinite- dimensional Hilbert spaces, is studied by using an implicit discretization scheme. It is assumed that the moving set depends on the time, the state and is possibly unbounded. The assumptions on the Lipschitz continuity and the compactness of the moving set, and the linear growth boundedness of the perturbation force are weaker than the ones used in previous papers.