Global stabilization of a chain of two integrators by a feedback in the form of nested sigmoids

The problem of stabilizing a chain of two integrators by a feedback in the form of nested sigmoids is considered. Such a feedback allows one to easily take into account boundedness of the control resource and ensure the fulfillment of desired characteristics of the transient process, such as a given exponential rate of the deviation decrease near the equilibrium state and the constraint on the maximum velocity. Global stability of the closed-loop system is proved by constructing its Lyapunov function.