  This work presents new tools for studying reachability and set invariance for
continuous-time mixed-monotone dynamical systems subject to a disturbance
input. The vector field of a mixed-monotone system is decomposable via a
decomposition function into increasing and decreasing components, and this
decomposition enables embedding the original dynamics in a higher-dimensional
embedding system. While the original system is subject to an unknown
disturbance input, the embedding system has no disturbances and its
trajectories provide bounds for finite-time reachable sets of the original
dynamics. Our main contribution is to show how one can efficiently identify
robustly forward invariant and attractive sets for mixed-monotone systems by
studying certain equilibria of this embedding system. We show also how this
approach, when applied to the backward-time dynamics, establishes different
robustly forward invariant sets for the original dynamics. Lastly, we present
an independent result for computing decomposition functions for systems with
polynomial dynamics. These tools and results are demonstrated through several
examples and a case study.
