Stabilization of a chain of three integrators subject to the control and phase constraints is discussed. The paper continues our previous work reported at the Optima 2022 conference, where a control law based on two nested saturators was proposed and was proved that it globally stabilizes the system in the lack of control and phase constraints. In this work, to meet the phase and control constraints, we apply an advanced six-parameter feedback law in the form of three nested saturation functions. The number of the parameters reduces to twothe control resource and the desired exponential convergence rate near the originby turning to dimensionless variables and taking three coefficients from a one-parameter family. The selection of the parameters is discussed that optimizes the performance of the controller. We define an optimal controller as that that globally stabilizes the system while ensuring the greatest convergence rate. The phase portrait of the closed-loop system is shown to be very complicated. When the convergence rate exceeds an optimal value, a bifurcation occurs resulting in an infinite number of periodic motions the trajectories of which lie on the surface of a cylinder in the phase space. A lower bound of the optimal convergence rate is obtained analytically, and its approximate value is found numerically.