For a class of uncertain nonlinear systems (UNSs), the
flow-invariance of a time-dependent rectangular set
(TDRS) defines individual constraints for each component of the
state-space trajectories. It is shown that the existence of the
flow-invariance property is equivalent to the existence of
positive solutions for some differential inequalities with
constant coefficients (derived from the state-space equation of
the UNS). Flow-invariance also provides basic tools for dealing
with the componentwise asymptotic stability as a special
type of asymptotic stability, where the evolution of the state
variables approaching the equilibrium point (EP){0} is separately monitored (unlike the standard asymptotic stability,
which relies on global information about the state variables,
formulated in terms of norms). The EP{0} of a given UNS is proved to be componentwise asymptotically stable if and only if the EP{0} of a differential equation with constant coefficients is asymptotically stable in the standard sense. Supplementary requirements for the individual evolution of the state variables approaching the EP{0} allow introducing the stronger concept of componentwise exponential asymptotic stability, which can be characterized by algebraic conditions. Connections with the componentwise asymptotic stability of an
uncertain linear system resulting from the linearization of a given UNS are also discussed.
