The synthesis of stabilizing saturated linear state feedback controllers is considered for both linear continuous and discrete time-invariant systems. The main idea is to stabilize the plant by a low gain linear state feedback first, then to synthesize another linear state feedback (based on a quadratic Lyapunov function) and to saturate the sum of the two controls. Sufficient conditions are found for stability of the closed-loop saturated system when the initial conditions are within a certain subset of the state space. Computer algorithms are presented for synthesis requiring interactive software for LQ design and simulation. An example of a submarine depth regulator is discussed.