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.