Monotonicity

Monotonicity refers to the property of a system whose trajectories preserve a partial order, which means that if we consider two ordered initial states (one “greater” than the other), the system trajectory from the first initial state will always remain “above” the trajectory from the second state. In this chapter, we particularly focus on partial orders induced by orthants of the Euclidean space and we present reachability methods taking advantage of this order-preserving property satisfied by monotone systems. When applicable, these methods are guaranteed to provide a tight interval over-approximation of the reachable set, for both continuous-time and discrete-time systems. In addition, this approach is particularly efficient and has a low computational complexity since the over-approximations are fully defined from the evaluation of only two successors of the system.