A controller design methodology is proposed for linear continuous-time systems with asymmetrical constrained control. Because of the trade-off between the size of the domain of attraction and convergence speed, one can improve the convergence speed during the system evolution. The closed-loop system starts with bad performances and a large domain of attraction and ends with good performances near the origin. The objective of this work is to overcome some discontinuities resulting from a piecewise linear control law previously proposed to improve the convergence speed of a linear system during its evolution. The proposed method is based on the use of the Sylvester equation together with a quadratic Lyapunov function. It guaranties the improvement of the speed of convergence without any discontinuities on the control vector components and respecting always the imposed asymmetric constraints without saturations.