# Computing Robustly Forward Invariant Sets for Mixed-Monotone Systems

arXiv: https://arxiv.org/abs/2003.05912 · HTML: https://mlsanity.com/paper/2003.05912/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Mar 2020.

## Built on by (most cited first, 3 of 3)

- [Robustness Certificates for Implicit Neural Networks: A Mixed Monotone Contractive Approach](https://mlsanity.com/paper/2112.05310/) (2021)
- [A Toolbox for Fast Interval Arithmetic in numpy with an Application to Formal Verification of Neural Network Controlled Systems](https://mlsanity.com/paper/2306.15340/) (2023)
- [Forward Invariance in Neural Network Controlled Systems](https://mlsanity.com/paper/2309.09043/) (2023)

## Abstract (arXiv)

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.