This paper presents a detailed study of the robust control of a cable’s vibrations, with emphasis on considering a model of infinite dimension. Indeed, using a partial differential equation model of the vibrations of an inclined cable with sag, we are interested in studying the application of H∞-robust feedback control to this infinite dimensional system. The approach relies on Riccati equations to stabilize the system under measurement feedback when it is subjected to external disturbances. Henceforth, our study focuses on the construction of a standard linear infinite dimensional state space description of the cable under consideration before writing its approximation of finite dimension and studying the H∞ feedback control of vibrations with partial observation of the state in both cases. The closed loop system is numerically simulated to illustrate the effectiveness of the resulting control law.