Tight Decomposition Functions for Continuous-Time Mixed-Monotone Systems With Disturbances

The vector field of a mixed-monotone system is decomposable via decomposition function into increasing (cooperative) and decreasing (competitive) components, this allows for, e.g., efficient computation reachable sets forward invariant sets. A main challenge in approach, however, identifying an appropriate function. In letter, we show that any continuous-time dynamical with Lipschitz continuous mixed-monotone, provide construction for the yields tightest approximation when used standard tools systems. Our similar to recently proposed by Yang Ozay computing functions discrete-time systems where make modifications setting also extend case unknown disturbance inputs. As Yang's Ozay's work, our requires solving optimization problem each point state-space; demonstrate through example how tight can sometimes be calculated closed form. second contribution, under-approximations efficiently computed mixed-monotonicity property considering backward-time dynamics.