Dynamics of a class of uncertain nonlinear systems under flow-invariance constraints

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.