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arxiv: 2303.17674 · v5 · submitted 2023-03-30 · 🧮 math.OC · cs.LG· cs.RO· cs.SY· eess.SY

Convex Hulls of Reachable Sets

Thomas Lew , Riccardo Bonalli , Marco Pavone This is my paper

Pith reviewed 2026-05-24 09:33 UTC · model grok-4.3

classification 🧮 math.OC cs.LGcs.ROcs.SYeess.SY
keywords reachable setsconvex hullsnonlinear systemsbounded disturbancessampling-based estimationmodel predictive controlneural feedback loops
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The pith

Convex hulls of reachable sets equal the convex hulls of ODE solutions started on the sphere.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper characterizes the convex hulls of reachable sets for nonlinear systems with bounded disturbances and uncertain initial conditions as the convex hulls of solutions to the corresponding ordinary differential equation with initial conditions on the sphere. This matters because direct computation of reachable sets is challenging, and existing methods are often conservative or expensive. The characterization provides a finite-dimensional description that supports an efficient sampling-based estimation algorithm. It further enables analysis of the boundary structure and derivation of error bounds for the estimates. The results are applied to neural feedback loop analysis and robust model predictive control.

Core claim

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.

What carries the argument

The ordinary differential equation solutions with initial conditions restricted to the sphere, whose convex hull yields the convex hull of the reachable set.

If this is right

  • Unlocks an efficient sampling-based estimation algorithm for over-approximating reachable sets.
  • Permits study of the boundary structure of the reachable convex hulls.
  • Derives error bounds for the sampling-based estimation algorithm.
  • Applies to analysis of neural feedback loops and design of robust model predictive controllers.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This approach could be tested on systems with specific nonlinearities to verify the reduction's accuracy.
  • The spherical initialization might connect to geometric properties in optimal control theory.
  • Extending the method to time-varying or hybrid systems could broaden its use in verification.

Load-bearing premise

The system dynamics are such that the convex hull of the reachable set is exactly the convex hull of the trajectories starting from the sphere without additional regularity conditions being required.

What would settle it

A specific nonlinear system with bounded disturbances and uncertain initials where some point in the convex hull of the reachable set cannot be expressed as a convex combination of ODE solutions from the sphere.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper studies convex hulls of reachable sets for nonlinear systems with bounded disturbances and uncertain initial conditions. It claims that these convex hulls equal the convex hulls of solutions to an ODE whose initial conditions lie on the sphere. This finite-dimensional reduction is used to derive a sampling-based over-approximation algorithm, boundary structure results, error bounds, and applications to neural feedback loop analysis and robust MPC.

Significance. If the central characterization holds under the necessary regularity assumptions, the reduction to spherical initial conditions would provide a non-conservative, finite-dimensional representation that enables efficient sampling-based estimation of reachable-set convex hulls. This would be a useful contribution to reachability analysis in control, where existing methods are often conservative or expensive, and the applications to neural feedback and robust MPC indicate practical relevance.

major comments (2)
  1. [Theorem 3.1 (main characterization)] Main characterization (Theorem 3.1 or equivalent statement of the spherical-initial-condition reduction): the equality conv(R) = conv{ solutions of the ODE with x(0) on the sphere } is stated without explicit hypotheses on f (e.g., local Lipschitz continuity in x uniformly in d) or on the disturbance set (compactness). Without these, uniqueness of solutions and attainment of the support function by extreme trajectories can fail, so the claimed identity does not hold in general.
  2. [Section 4] Section 4 (sampling algorithm and error bounds): the error bounds for the sampling-based estimator rely on the convex-hull characterization being exact; if the regularity conditions are omitted from the theorem, the bounds do not apply to the systems described in the abstract.
minor comments (2)
  1. [Section 2] Notation for the reachable set R and the disturbance set should be introduced with explicit definitions before the main theorem.
  2. [Introduction / Section 2] The abstract mentions 'uncertain initial conditions' but the precise set of initial conditions (e.g., a ball or ellipsoid) is not stated until later; move this to the problem formulation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the opportunity to respond to the referee's report. We address the major comments point by point below. We agree that the regularity assumptions should be stated more explicitly in the theorem and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Theorem 3.1 (main characterization)] Main characterization (Theorem 3.1 or equivalent statement of the spherical-initial-condition reduction): the equality conv(R) = conv{ solutions of the ODE with x(0) on the sphere } is stated without explicit hypotheses on f (e.g., local Lipschitz continuity in x uniformly in d) or on the disturbance set (compactness). Without these, uniqueness of solutions and attainment of the support function by extreme trajectories can fail, so the claimed identity does not hold in general.

    Authors: We thank the referee for pointing this out. The problem setup in Section 2 assumes that f is continuous and locally Lipschitz continuous in the state variable x, uniformly with respect to the disturbance d, and that the disturbance set is compact. These conditions ensure the existence and uniqueness of solutions to the ODE. However, we acknowledge that these hypotheses are not restated explicitly in the statement of Theorem 3.1. We will revise the theorem statement to include these assumptions explicitly, thereby ensuring the claimed equality holds under the stated conditions. revision: yes

  2. Referee: [Section 4] Section 4 (sampling algorithm and error bounds): the error bounds for the sampling-based estimator rely on the convex-hull characterization being exact; if the regularity conditions are omitted from the theorem, the bounds do not apply to the systems described in the abstract.

    Authors: The error bounds presented in Section 4 are derived based on the characterization provided by Theorem 3.1. By explicitly incorporating the necessary regularity assumptions into Theorem 3.1 as described in our response to the first comment, the error bounds will be applicable to the class of systems considered in the paper, including those in the abstract. We will also add a clarifying statement in Section 4 to reference the assumptions from Theorem 3.1. revision: yes

Circularity Check

0 steps flagged

No circularity: characterization derived from reachable-set properties

full rationale

The paper claims a characterization of conv(R) as the convex hull of ODE trajectories starting on the sphere. This is presented as a derived identity from the definition of reachable sets under bounded disturbances, without any reduction to a fitted parameter, self-citation chain, or input renamed as output. No load-bearing self-citations or ansatzes are referenced in the abstract or reader summary; the result is a standard first-principles consequence of support-function or extreme-point arguments for differential inclusions, making the derivation self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract.

pith-pipeline@v0.9.0 · 5650 in / 960 out tokens · 22597 ms · 2026-05-24T09:33:16.477404+00:00 · methodology

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Reference graph

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    Proof: The Gauss map n : ∂C → S n−1, x 7→ ∇h(x)/∥∇h(x)∥ is well-defined under the assumptions

    because their level set functions h(x) behave approx- imately like the squared norm function h(x) = ∥x∥2. Proof: The Gauss map n : ∂C → S n−1, x 7→ ∇h(x)/∥∇h(x)∥ is well-defined under the assumptions. Thus, the map ∇h and its inverse are well-defined. First, n(n−1(d)) = ∇h(n−1(d))/∥∇h(n−1(d))∥ (7) = d. for all d ∈ S n−1. Second, for any x ∈ ∂C , n−1(n(x))...

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    Then, ∇h(x) = 2 Q−1(x − ¯x), so ∇h−1(y) = ¯x + 1 2 Qy and n(x) = Q−1(x−¯x) ∥Q−1(x−¯x)∥

    Ellipsoid: Let h(x) = ( x − ¯x)⊤Q−1(x − ¯x), so C = E. Then, ∇h(x) = 2 Q−1(x − ¯x), so ∇h−1(y) = ¯x + 1 2 Qy and n(x) = Q−1(x−¯x) ∥Q−1(x−¯x)∥. Then, ∇h−1(n(x)) = ¯x + 1 2 (x−¯x) ∥Q−1(x−¯x)∥, so that h(∇h−1(n(x))) = 1 4 (x − ¯x)⊤Q−1(x − ¯x) ∥Q−1(x − ¯x)∥2 = 1 4 1 ∥Q−1(x − ¯x)∥2 = 1 ∥∇h(x)∥2 , as h(x) = 1 for all x ∈ ∂C . Thus, (65) is verified

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    Then, ∇h(x) = 2 (x−¯x)⊙|x−¯x|λ−2⊙δ¯x−λ ∥(x−¯x)⊙δ¯x−1∥λ−2 λ , so ∇h−1(y) = ¯ x + 1 2 ∥|y ⊙ δ¯x| 1 λ−1 ∥λ−2 λ y 1 λ−1 ⊙ δ¯x λ λ−1

    λ-balls: For λ > 1, let h(x) = ∥x − ¯x ⊙ δ¯x−1∥2 λ, so C = (43a). Then, ∇h(x) = 2 (x−¯x)⊙|x−¯x|λ−2⊙δ¯x−λ ∥(x−¯x)⊙δ¯x−1∥λ−2 λ , so ∇h−1(y) = ¯ x + 1 2 ∥|y ⊙ δ¯x| 1 λ−1 ∥λ−2 λ y 1 λ−1 ⊙ δ¯x λ λ−1 . Then, with n(x) = (x − ¯x) ⊙ |x − ¯x|λ−2 ⊙ δ¯x−λ ∥|x − ¯x|λ−1 ⊙ δ¯x−λ∥ , (44) one verifies that ∇h−1(n(x))) = ¯x + 1 2 ∥(x − ¯x) ⊙ δ¯x−1∥λ−2 ∥|x − ¯x|λ−1 ⊙ δ¯x−λ...

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    The initial direction values d0 for Algorithm 1 are selected to evenly cover the circle S n−1. F . Details on the spacecraft attitude control results Implementation: We use T = 10s, discretize (61) as x((k + 1)∆t) = ¯f(x(k∆t), ¯u(k∆t)) + w(k∆t), (75) 7Note that by selecting appropriate hyperparameters, Softplus activations can be made arbitrarily close to...

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