# Tight Decomposition Functions for Continuous-Time Mixed-Monotone Systems with Disturbances

arXiv: https://arxiv.org/abs/2003.07975 · HTML: https://mlsanity.com/paper/2003.07975/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Mar 2020.

## Built on by (most cited first, 4 of 4)

- [Interval Reachability of Nonlinear Dynamical Systems with Neural Network Controllers](https://mlsanity.com/paper/2301.07912/) (2023)
- [Efficient Interaction-Aware Interval Analysis of Neural Network Feedback Loops](https://mlsanity.com/paper/2307.14938/) (2023)
- [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)

## Abstract (arXiv)

The vector field of a mixed-monotone system is decomposable via a decomposition function into increasing (cooperative) and decreasing (competitive) components, and this decomposition allows for, e.g., efficient computation of reachable sets and forward invariant sets. A main challenge in this approach, however, is identifying an appropriate decomposition function. In this work, we show that any continuous-time dynamical system with a Lipschitz continuous vector field is mixed-monotone, and we provide a construction for the decomposition function that yields the tightest approximation of reachable sets when used with the standard tools for mixed-monotone systems. Our construction is similar to that recently proposed by Yang and Ozay for computing decomposition functions of discrete-time systems [1] where we make appropriate modifications for the continuous-time setting and also extend to the case with unknown disturbance inputs. As in [1], our decomposition function construction requires solving an optimization problem for each point in the state-space; however, we demonstrate through example how tight decomposition functions can sometimes be calculated in closed form. As a second contribution, we show how under-approximations of reachable sets can be efficiently computed via the mixed-monotonicity property by considering the backward-time dynamics.