# Scalable Zonotopic Under-approximation of Backward Reachable Sets for Uncertain Linear Systems

arXiv: https://arxiv.org/abs/2107.01724 · HTML: https://mlsanity.com/paper/2107.01724/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Jul 2021.

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

- [Backward Reachability Analysis for Neural Feedback Loops](https://mlsanity.com/paper/2204.08319/) (2022)
- [Provably-Safe Neural Network Training Using Hybrid Zonotope Reachability Analysis](https://mlsanity.com/paper/2501.13023/) (2025)
- [Towards Safe Reinforcement Learning with Reduced Conservativeness: A Case Study on Drone Flight Control](https://mlsanity.com/paper/2608.26852/) (2026)

## Abstract (arXiv)

Zonotopes are widely used for over-approximating forward reachable sets of uncertain linear systems for verification purposes. In this paper, we use zonotopes to achieve more scalable algorithms that under-approximate backward reachable sets of uncertain linear systems for control design. The main difference is that the backward reachability analysis is a two-player game and involves Minkowski difference operations, but zonotopes are not closed under such operations. We under-approximate this Minkowski difference with a zonotope, which can be obtained by solving a linear optimization problem. We further develop an efficient zonotope order reduction technique to bound the complexity of the obtained zonotopic under-approximations. The proposed approach is evaluated against existing approaches using randomly generated instances and illustrated with several examples.