Positive invariance, monotonicity and comparison of nonlinear systems

In this paper, the interrelation between positive invariance, monotonicity and comparison of iterated nonlinear systems defined in partially ordered sets is studied. First, necessary and sufficient conditions guaranteeing the positive invariance of sets defined by relations of the form v(x)≤w with respect to nonlinear systems are established. Then, various characterizations of monotone nonlinear systems are developed and a connection between positive invariance and monotonicity is established. Finally, necessary and sufficient conditions for a, not necessarily monotone, nonlinear system to be a comparison system are established. No specific algebraic structure is necessary for developing the main results of this work. Numerical examples illustrating the applicability of all these results to the case of discrete-time dynamical systems are also given.