  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.
