Guaranteeing in-polytope permanence times of nonlinear system trajectories via approximate solutions

Causing the system trajectory to remain within a given region over some time interval is a key requirement of many control strategies. This may involve prediction of the future state evolution. However, general nonlinear systems often lack analytical solutions, making numerical approximations the only available tool. Even if error bounds for approximate solutions exist, these only ensure that the exact solution remains within the given region at discrete time instants. This fact may nonetheless not be sufficient to guarantee that the exact continuous-time solution remains within the region at all intermediate times. In this work, we develop a method to guarantee that the exact solution remains within a convex set throughout a time interval, by employing approximate solutions. When the convex set is a polytope, every step of the method becomes easily and completely computable. The applicability of the method is illustrated by means of numerical examples.