Convex Hulls of Reachable Sets

arXiv cs.RO / 4/16/2026

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Key Points

The paper studies how to compute or approximate the convex hulls of reachable sets for nonlinear control systems with bounded disturbances and uncertain initial conditions.
It provides a finite-dimensional characterization, showing these convex hulls can be expressed as convex hulls of solutions of an ordinary differential equation with initial conditions constrained to a sphere.
This characterization enables an efficient sampling-based algorithm to produce tighter (less conservative) over-approximations of reachable sets.
The work analyzes the geometry of the boundary of the reachable convex hulls and derives error bounds for the estimation method.
It demonstrates applications in neural feedback loop analysis and robust MPC, indicating practical relevance for modern control workflows using learning-based components.

Abstract

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.