# Decomposition of Reachable Sets and Tubes for a Class of Nonlinear Systems

arXiv: https://arxiv.org/abs/1611.00122 · HTML: https://mlsanity.com/paper/1611.00122/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Nov 2016.

## Built on by (most cited first, 18 of 18)

- [The Safety Filter: A Unified View of Safety-Critical Control in Autonomous Systems](https://mlsanity.com/paper/2309.05837/) (2023)
- [Iterative Reachability Estimation for Safe Reinforcement Learning](https://mlsanity.com/paper/2309.13528/) (2023)
- [Hamilton-Jacobi Reachability in Reinforcement Learning: A Survey](https://mlsanity.com/paper/2407.09645/) (2024)
- [Scalable Learning of Safety Guarantees for Autonomous Systems using Hamilton-Jacobi Reachability](https://mlsanity.com/paper/2101.05916/) (2021)
- [Safe Autonomous Racing via Approximate Reachability on Ego-vision](https://mlsanity.com/paper/2110.07699/) (2021)
- [Risk-averse autonomous systems: A brief history and recent developments from the perspective of optimal control](https://mlsanity.com/paper/2109.08947/) (2021)
- [Infinite-Horizon Reach-Avoid Zero-Sum Games via Deep Reinforcement Learning](https://mlsanity.com/paper/2203.10142/) (2022)
- [Decomposing Control Lyapunov Functions for Efficient Reinforcement Learning](https://mlsanity.com/paper/2403.12210/) (2024)
- [TTT: A Temporal Refinement Heuristic for Tenuously Tractable Discrete Time Reachability Problems](https://mlsanity.com/paper/2407.14394/) (2024)
- [Learning Maximal Safe Sets Using Hypernetworks for MPC-based Local Trajectory Planning in Unknown Environments](https://mlsanity.com/paper/2410.20267/) (2024)
- [Residual Neural Terminal Constraint for MPC-based Collision Avoidance in Dynamic Environments](https://mlsanity.com/paper/2508.03428/) (2025)
- [Certified Approximate Reachability (CARe): Formal Error Bounds on Deep Learning of Reachable Sets](https://mlsanity.com/paper/2503.23912/) (2025)
- [Verification of safety critical control policies using kernel methods](https://mlsanity.com/paper/2203.12407/) (2022)
- [Certifying Hamilton-Jacobi Reachability Learned via Reinforcement Learning](https://mlsanity.com/paper/2602.16475/) (2026)
- [Physical Backdoor Trigger Activation of Autonomous Vehicle using Reachability Analysis](https://mlsanity.com/paper/2303.13992/) (2023)
- [Reach-Avoid Differential game with Reachability Analysis for UAVs: A decomposition approach](https://mlsanity.com/paper/2512.22793/) (2025)
- [Unifying Hamilton-Jacobi Reachability and Reinforcement Learning](https://mlsanity.com/paper/2601.08050/) (2026)
- [V-MORALS: Visual Morse Graph-Aided Estimation of Regions of Attraction in a Learned Latent Space](https://mlsanity.com/paper/2602.23524/) (2026)

## Abstract (arXiv)

Reachability analysis provides formal guarantees for performance and safety properties of nonlinear control systems. Here, one aims to compute the backward reachable set (BRS) or tube (BRT) -- the set of states from which the system can be driven into a target set at a particular time or within a time interval, respectively. The computational complexity of current approaches scales exponentially, making application to high-dimensional systems intractable. We propose a technique that decomposes the dynamics of a general class of nonlinear systems into subsystems which may be coupled through common states, controls, and disturbances. Despite this coupling, BRSs and BRTs can be computed efficiently using our technique without incurring additional approximation errors and without the need for linearizing dynamics or approximating sets as polytopes. Computations of BRSs and BRTs now become orders of magnitude faster, and for the first time BRSs and BRTs for many high-dimensional nonlinear control systems can be computed using the Hamilton-Jacobi (HJ) formulation. In situations involving bounded adversarial disturbances, our proposed method can obtain slightly conservative results. We demonstrate our theory by numerically computing BRSs and BRTs using the HJ formulation for several systems, including the 6D Acrobatic Quadrotor and the 10D Near-Hover Quadrotor.