  This letter presents a unified framework for the design of prescribed-time
controllers under time-varying input and state constraints for normal-form
unknown nonlinear systems with uncertain input gain. The proposed approach is
based on a time-domain mapping method by which any infinite-time system can be
corresponded to a prescribed-time system and vice versa. It is shown that the
design of a constrained nonasymptotic prescribed-time controller can be reduced
to the asymptotic control design for an associated constrained infinite-time
system. Fa\`a di Bruno's formula and Bell polynomials are used for a
constructive representation of the associated infinite-time system. The
presented results are not confined to a particular mapping function, which adds
to the flexibility of the proposed scheme. It is shown that necessary and
sufficient conditions on the uniform (practical) prescribed-time stability and
attractivity can be obtained as corollaries of the main result.
