Generalization of a flow-invariance criterion

The paper provides new results for testing the flow-invariance of time-dependent sets with arbitrary shapes, with respect to trajectories of continuous-time nonlinear systems. The approach represents a generalization of a criterion used in the qualitative analysis of ordinary differential equations with solutions in closed sets with hyper-rectangular form. Given a nonlinear dynamical system for investigation, our generalization consists in the use of an auxiliary system with pseudo-linear form, whose trajectories play a comparison role for the trajectories of the investigated system. The various shapes of the invariant set candidates can be defined in terms of Hölder p-norms of some smooth time-dependent vector functions. The non-symmetrical and symmetrical invariant sets are addressed separately, since the construction of the auxiliary systems requires different properties for their trajectories. The developments we propose are able to include, as particular cases, several earlier studies on flow-invariance, which were separately reported in literature for different classes of dynamics.