The problem of stabilizing the chain of three integrators by a piecewise continuous constrained control is studied. A feedback law in the form of three nested saturators specified by six—three model and three design—parameters is proposed. Global stability of the closed-loop system is studied, and an optimization problem of determining the feedback coefficients ensuring the greatest convergence rate near the equilibrium while preserving global asymptotic stability is stated. It is shown that the loss of global stability results from arising hidden attractors, which come to existence when the convergence rate becomes greater than or equal to a critical value depending on the control resource. A numerical procedure for constructing hidden attractors is developed. The bifurcation value of the convergence rate, which is an exact upper bound of the parameter values ensuring global asymptotic stability of the closed-loop system, is determined numerically by solving an algebraic system of four equations.