A stabilizing controller design for linear systems subject to asymmetric actuator saturation is proposed. Stabilization is achieved by focusing on shifted equilibria selected via the solution of an optimization problem ensuring convergence to the origin of the shifted equilibrium. To enable the computational time required by the optimizer, we impose sampled-data updates of the shifted equilibria and cast our description within a hybrid dynamical systems formulation. Two feedback solutions are given, using exact and inaccurate optimization algorithms, thus establishing interesting trade-offs between continuous-time dynamics and computationally expensive iterative discrete-time parametric optimization schemes. Through numerical examples, estimates of the region of attraction obtained through the method outlined in this paper are compared to other methods in the literature. Additionally, the real-time applicability of the control law is illustrated on numerical examples.