# Convex Hulls of Reachable Sets

arXiv: https://arxiv.org/abs/2303.17674 · HTML: https://mlsanity.com/paper/2303.17674/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 80 references that are in the map.
- First posted on arXiv 30 Mar 2023.

## Builds on (most cited first, 11 of 11)

- [Learning-based Model Predictive Control for Safe Exploration](https://mlsanity.com/paper/1803.08287/) (2018)
- [Reachability Analysis of Neural Feedback Loops](https://mlsanity.com/paper/2108.04140/) (2021)
- [TIRA: Toolbox for Interval Reachability Analysis](https://mlsanity.com/paper/1902.05204/) (2019)
- [A Simple and Efficient Sampling-based Algorithm for General Reachability Analysis](https://mlsanity.com/paper/2112.05745/) (2021)
- [Sampling-based Reachability Analysis: A Random Set Theory Approach with Adversarial Sampling](https://mlsanity.com/paper/2008.10180/) (2020)
- [Robust Nonlinear Optimal Control via System Level Synthesis](https://mlsanity.com/paper/2301.04943/) (2023)
- [Sequential Convex Programming For Non-Linear Stochastic Optimal Control](https://mlsanity.com/paper/2009.05182/) (2020)
- [Learning Approximate Forward Reachable Sets Using Separating Kernels](https://mlsanity.com/paper/2011.09678/) (2020)
- [Analysis of Theoretical and Numerical Properties of Sequential Convex Programming for Continuous-Time Optimal Control](https://mlsanity.com/paper/2009.05038/) (2020)
- [Ellipsotopes: Combining Ellipsoids and Zonotopes for Reachability Analysis and Fault Detection](https://mlsanity.com/paper/2108.01750/) (2021)
- [Estimating the Convex Hull of the Image of a Set with Smooth Boundary: Error Bounds and Applications](https://mlsanity.com/paper/2302.13970/) (2023)

## Abstract (arXiv)

We study the convex hulls of reachable sets of nonlinear systems with bounded disturbances and uncertain initial conditions. Reachable sets play a critical role in control, but remain notoriously challenging to compute, and existing over-approximation tools tend to be conservative or computationally expensive. In this work, we characterize the convex hulls of reachable sets as the convex hulls of solutions of an ordinary differential equation with initial conditions on the sphere. This finite-dimensional characterization unlocks an efficient sampling-based estimation algorithm to accurately over-approximate reachable sets. We also study the structure of the boundary of the reachable convex hulls and derive error bounds for the estimation algorithm. We give applications to neural feedback loop analysis and robust MPC.