  This paper revisits a classical challenge in the design of stabilizing
controllers for nonlinear systems with a norm-bounded input constraint. By
extending Lin-Sontag's universal formula and introducing a generic
(state-dependent) scaling term, a unifying controller design method is
proposed. The incorporation of this generic scaling term gives a unified
controller and enables the derivation of alternative universal formulas with
various favorable properties, which makes it suitable for tailored control
designs to meet specific requirements and provides versatility across different
control scenarios. Additionally, we present a constructive approach to
determine the optimal scaling term, leading to an explicit solution to an
optimization problem, named optimization-based universal formula. The resulting
controller ensures asymptotic stability, satisfies a norm-bounded input
constraint, and optimizes a predefined cost function. Finally, the essential
properties of the unified controllers are analyzed, including smoothness,
continuity at the origin, stability margin, and inverse optimality. Simulations
validate the approach, showcasing its effectiveness in addressing a challenging
stabilizing control problem of a nonlinear system.
