Tight decomposition functions for mixed monotonicity

Mixed monotonicity is a property of a system's vector field that says that the vector field admits a decomposition function, in a lifted space, that has some order-preserving properties. It is recently shown that this property allows one to efficiently over-approximate the system's one-step reachable set with a hyperinterval, which is obtained by evaluating the vector field's decomposition function at two points. Such decomposition functions are usually not unique and some decompositions may not be tight in the sense that the resulting hyperintervals are not the smallest ones that contain the exact one-step reachable set, which leads to conservative over-approximation. In this paper, we show that for a general class of functions, there exists a tight decomposition, which can be implicitly constructed as the solution of certain optimization problems. This implies that any function from ℝⁿ to ℝ^m (hence any forward complete system) is mixed-monotone. However, the usefulness of the constructed tight decomposition functions is limited by the fact that it might not be possible to evaluate them efficiently. We show that under certain conditions, the tight decompositions can reduce to a function with explicit expression, which can be directly evaluated. This result suggests that it is not mixed monotonicity itself, but other extra properties, which lead to explicitly evaluatable decomposition functions, that enable efficient and tight hyperinterval over-approximation of reachable sets.