Convex Hulls of Reachable Sets

TL;DR

The paper proves a finite-dimensional, convex-geometry–driven characterization of the convex hulls of reachable sets for nonlinear systems with bounded disturbances and uncertain initial conditions: $\textrm{H}(\mathcal{X}_t)=\textrm{H}(F(\mathcal{S}^{n-1},t))$, where $F$ is the solution map of an augmented ODE with initial data on $\mathcal{S}^{n-1}$. This leads to an efficient sampling-based estimator (Algorithm 1) that reconstructs reachable hulls by integrating the PMPODE from a finite set of directions on the unit sphere, with provable error bounds on the estimation. The framework extends to rectangular uncertainty sets via smooth $\lambda$-norm approximations and to non-invertible $g$ through full-rank extensions, providing convergent inner/outer hull estimates. The approach is validated on neural feedback loops, Dubins-car scenarios, and spacecraft attitude control with robust MPC, demonstrating tighter, less-conservative hulls and competitive computation times compared with baseline reachability methods. Overall, the work offers a scalable, geometry-grounded method to certify and optimize robustness in complex nonlinear control systems.

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.

Figures (12)

• Figure 1: The convex hulls $\textrm{H}(\mathcal{X}_t)$ of the reachable sets $\mathcal{X}_t$ can be computed by (a) integrating an augmented ODE (\ref{['PMPODE']}) for different directions $d^0$ on the sphere $\mathcal{S}^{n-1}$, and (b) taking the convex hulls of the resulting extremal state trajectories $x_{d^0}$.
• Figure 2: Gauss map $n^\mathcal{M}:\mathcal{M}\to\mathcal{S}^{n-1}$ of an ovaloid $\mathcal{M}=\partial\mathcal{C}$.
• Figure 3: Solutions of \ref{['PMPODE']} for all directions $d^0\in\mathcal{S}^{n-1}$ at different times $t\in[0,T]$ for the attraction-repulsion system in Example \ref{['example:selfintersect']}.
• Figure 4: Two support hyperplanes at $x{=}x_w(T)$.
• Figure 5: Smooth under- and over-approximations of a rectangular set $C$ for different relaxation parameters $\lambda$. As $\lambda$ increases, the approximations converge to the set $C$.

Theorems & Definitions (25)

• Lemma 1: Support hyperplane
• Lemma 2
• Lemma 3: $\mathcal{X}_t$ is compact
• Corollary 1: Reachable tube
• Lemma 4: No singular arcs
• proof
• Lemma 5: Extremals of \ref{['OCP']} are identified by $\frac{p(0)}{\|p(0)\|}$
• proof
• Lemma 6
• proof