Flow invariance in exploring stability for a class of nonlinear uncertain systems

For a class of nonlinear uncertain systems, the flow-invariance of time-dependent hyper-rectangles is introduced as a concept of geometric nature, defining 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 equations of the nonlinear uncertain systems). Flow-invariance offers instruments for defining and studying the componentwise (exponential) asymptotic stability, as a special type of (exponential) asymptotic stability, where the evolution of the state variables approaching the equilibrium point {0} is separately monitored (unlike the standard asymptotic stability, which relies on global information about the state variables, formulated in terms of norms). Connections with the componentwise asymptotic stability of a linear uncertain system, which result from the linearization of a nonlinear uncertain system belonging to the explored class, are also discussed.