# Over- and Under-Approximating Reachable Sets for Perturbed Delay Differential Equations

arXiv: https://arxiv.org/abs/1812.11718 · HTML: https://mlsanity.com/paper/1812.11718/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Dec 2018.

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

- [Safety Verification for Neural Networks Based on Set-boundary Analysis](https://mlsanity.com/paper/2210.04175/) (2022)
- [Mixed Monotonicity Reachability Analysis of Neural ODE: A Trade-Off Between Tightness and Efficiency](https://mlsanity.com/paper/2510.17859/) (2025)
- [Verifying Safety of Neural Networks from Topological Perspectives](https://mlsanity.com/paper/2306.15403/) (2023)
- [Inverse Delayed Reinforcement Learning](https://mlsanity.com/paper/2412.02931/) (2024)

## Abstract (arXiv)

This note explores reach set computations for perturbed delay differential equations (DDEs). The perturbed DDEs of interest in this note is a class of DDEs whose dynamics are subject to perturbations, and their solutions feature the local homeomorphism property with respect to initial states. Membership in this class of perturbed DDEs is determined by conducting sensitivity analysis of solution mappings with respect to initial states to impose a bound constraint on the time-lag term. The homeomorphism property of solutions to such class of perturbed DDEs enables us to construct over- and under-approximations of reach sets by performing reachability analysis on just the boundaries of their permitted initial sets, thereby permitting an extension of reach set computation methods for ordinary differential equations to perturbed DDEs. Three examples demonstrate the performance of our approach.