A linear continuous-time system with saturation on controls becomes a non-linear closed-loop system when linear state feedback control law is implemented. Assuming that the system to be controlled is stabilizable, the possibilities for an invariant feedback matrix to make the saturated closed-loop system globally asymptotically stable (GAS) are studied. It is proven that this stability property can be obtained by means of a linear feedback provided the considered system is stabilizable with respect to a Lyapunov function of the open-loop system. It is also shown that the global stability property in a closed-loop can only concern asymptotically or critically stable open-loop systems. However, except for a strictly unstable linear open-loop system, i.e. with at least one eigenvalue with a positive real part, there always exists some linear feedback linear matrix for which the stability domain can be as large as desired, even if the open-loop system is critically unstable. In all previous cases, the suitable feedback matrix is given and can easily be computed. When the closed-loop system cannot be GAS, a simple determination of a non-linear stability domain is given, together with the corresponding algorithm. Further, when the saturated closed-loop system is GAS, the conditions indicating whether this system is dynamically faster (DF) or equivalent (DE) to the open-loop system are given.