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arxiv: 2505.24744 · v3 · submitted 2025-05-30 · 🧮 math.OC

Universal Formulas for Safe Control and Their Neural Network Approximations

Pol Mestres , Jorge Cort\'es , Eduardo D. Sontag This is my paper

Pith reviewed 2026-05-19 12:47 UTC · model grok-4.3

classification 🧮 math.OC
keywords safe controlneural network controllersaffine inequalitiesuniversal formulasconvex function minimizationquadratic programming approximationinput constraints
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The pith

A neural network approximates smooth controllers that satisfy arbitrary affine inequalities independently of state dimension.

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

The paper studies how to design controllers that obey any number of affine inequalities at every state to guarantee safety, stability, and input limits. It defines the controller as the minimizer of a strictly convex function that enforces these inequalities automatically. A neural network then approximates this minimizer. The network serves as a universal formula for all tasks with input dimension and constraint number below given limits, works for any state dimension, and needs training data only from a bounded state-space region.

Core claim

The controller at each state is the minimizer of a strictly convex function. This minimizer can be approximated arbitrarily closely by a neural network. The network depends only on the input dimension and the number of affine inequalities, not on the state dimension. Training requires data from only a bounded set because the approximation holds uniformly on compact sets.

What carries the argument

Minimizer of a strictly convex function whose value satisfies the affine inequalities by construction, approximated by a neural network independent of state dimension.

If this is right

  • The resulting controller is smooth rather than piecewise, potentially improving closed-loop performance.
  • Real-time evaluation requires only a neural network inference instead of solving a quadratic program online.
  • One trained network applies to any system with the same input dimension and constraint count.
  • Data collection for training is restricted to a compact region, simplifying the process for high-dimensional states.

Where Pith is reading between the lines

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

  • This approach could be combined with learned models of system dynamics for end-to-end safe control synthesis.
  • Similar universal approximators might be developed for other types of constraints beyond affine inequalities.
  • Deployment in embedded systems becomes feasible where computational resources prohibit online optimization.

Load-bearing premise

The minimizer of the selected strictly convex function satisfies all the affine inequalities at every state in the domain.

What would settle it

A state and input where the neural network output violates one of the affine inequalities, even as the network approximation accuracy increases.

Figures

Figures reproduced from arXiv: 2505.24744 by Eduardo D. Sontag, Jorge Cort\'es, Pol Mestres.

Figure 2
Figure 2. Figure 2: (top) Evolution of min i∈[9] hi(x) along different tra￾jectories induced by the NN-based controller from different initial conditions (IC). (bottom) Evolution of V (x) along trajectories induced by the NN-based controller. ogous to that of [13, Lemma 5.2], the inequalities {ai(x) +bi(x) ⊤u < 0} 10 i=1, are simultaneously feasi￾ble at all points where the vectors {x − ci} 9 i=1 and x are linearly independen… view at source ↗
Figure 1
Figure 1. Figure 1: (left) Trajectories of the closed-loop system ob [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 4
Figure 4. Figure 4: (left) Projection in the (x, y) plane of trajectories of the closed-loop system obtained from the neural network based controller for Example 5.2. (right) Projection in the (x, y) plane of trajectories of the closed-loop system obtained from numerically finding the controller u ∗ online and warm￾starting the solver with the NN-based controller for Exam￾ple 5.2. Initial conditions are denoted by red crosses… view at source ↗
read the original abstract

We study the problem of designing a controller that satisfies an arbitrary number of affine inequalities at every point in the state space. This is motivated by the fact that a variety of key control objectives, such as stability, safety, and input saturation, are guaranteed by closed-loop systems whose controllers satisfy such inequalities. Many works in the literature design such controllers as the solution to a state-dependent quadratic program (QP) whose constraints are precisely the inequalities. When the input dimension and number of constraints are high, computing a solution of this QP in real time can become computationally burdensome. Additionally, the solution of such optimization problems is not smooth in general, which can degrade the performance of the system. This paper provides a novel method to design a smooth controller that satisfies an arbitrary number of affine constraints. The controller is given at every state as the minimizer of a strictly convex function. To avoid computing the minimizer of such function in real time, we introduce a method based on neural networks (NN) to approximate the controller. Remarkably, this NN can be used to solve the controller design problem for any task with less than a fixed input dimension and number of affine constraints, and is completely independent of the state dimension. This is why we refer to such NN approximation as a NN-based universal formula for control. Additionally, we show that the NN-based controller only needs to be trained with datapoints from a bounded set in the state space, which significantly simplifies the training process.

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 proposes designing smooth controllers satisfying arbitrary affine state-dependent inequalities by taking the minimizer of a strictly convex function at each state. It then approximates this minimizer via a neural network whose input dimension depends only on the input dimension m and number of constraints k (not on state dimension n), claims this NN is universal across all tasks with at most those m and k, and asserts that training data generated from states in a bounded set suffices for the approximation to work on arbitrary tasks.

Significance. If the universality and bounded-training claims hold with rigorous error bounds that guarantee constraint satisfaction, the result would enable reusable, smooth, real-time controllers for high-dimensional systems with many safety or saturation constraints, bypassing per-task QP solves and state-space gridding.

major comments (2)
  1. [Neural Network Approximation / Controller Construction] The central universality claim (abstract and NN-approximation section) requires that a single NN trained on bounded-state samples generalizes to arbitrary continuous maps A(x), b(x) arising from any task. The skeptic note correctly identifies that continuity of the argmin map alone does not guarantee the image of a bounded ball under (A,b) is dense in the compact set of admissible (A,b) pairs; without an explicit homogeneity, normalization, or covering argument that maps all possible constraint coefficients into the trained region, constraint violation can occur for unseen tasks.
  2. [Controller Construction] § Controller Construction (implied by the problem statement and weakest-assumption note): the claim that the minimizer of the chosen strictly convex function satisfies all given affine inequalities at every state is asserted but not derived from first principles. An explicit verification that the chosen objective forces the minimizer into the feasible set for arbitrary A(x), b(x) is load-bearing for both the exact controller and its NN approximation.
minor comments (2)
  1. Notation for the strictly convex function and its parameters should be introduced with explicit dependence on m and k to clarify the fixed-input-dimension claim.
  2. The manuscript would benefit from a short table comparing computational cost and smoothness of the proposed NN controller versus standard QP solvers for increasing n, m, k.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable feedback on our manuscript. We address each of the major comments below and will make revisions to improve the clarity and rigor of the presentation.

read point-by-point responses

  1. Referee: The central universality claim (abstract and NN-approximation section) requires that a single NN trained on bounded-state samples generalizes to arbitrary continuous maps A(x), b(x) arising from any task. The skeptic note correctly identifies that continuity of the argmin map alone does not guarantee the image of a bounded ball under (A,b) is dense in the compact set of admissible (A,b) pairs; without an explicit homogeneity, normalization, or covering argument that maps all possible constraint coefficients into the trained region, constraint violation can occur for unseen tasks.

    Authors: We appreciate this insightful observation on the universality claim. The manuscript relies on the continuity of the argmin map and the fact that the NN input depends solely on m and k. To address the concern, we will revise the NN-approximation section to include an explicit normalization of the constraint pairs (A, b) and a covering argument demonstrating that samples from bounded states, when appropriately scaled, can densely approximate the admissible set. This ensures the approximation error bounds hold uniformly for arbitrary tasks, preventing constraint violations. revision: yes

  2. Referee: the claim that the minimizer of the chosen strictly convex function satisfies all given affine inequalities at every state is asserted but not derived from first principles. An explicit verification that the chosen objective forces the minimizer into the feasible set for arbitrary A(x), b(x) is load-bearing for both the exact controller and its NN approximation.

    Authors: We agree that an explicit derivation is necessary for rigor. In the revised manuscript, we will add a detailed proof in the Controller Construction section showing from first principles that the minimizer of the strictly convex function satisfies the affine inequalities. This will involve analyzing the KKT conditions or the properties of the objective function that penalize violations in a way that forces feasibility at the optimum for any A(x) and b(x). revision: yes

Circularity Check

0 steps flagged

Standard convex optimization and NN approximation with minor self-citation

full rationale

The paper's derivation relies on established results from convex optimization for the existence of the minimizer and universal approximation theorems for neural networks. The universality claim for fixed input dimension (m and k) independent of state dimension n is supported by the structure of the QP and continuity arguments, without reducing to self-defined fitted parameters or load-bearing self-citations that lack independent verification. Training on bounded state sets is justified by compactness and continuity of the argmin map, not by construction from the same data. This aligns with a low circularity score as the central claims have independent mathematical content.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard convex-analysis results and the existence of a suitable strictly convex function whose minimizer encodes the inequalities; no new entities are postulated.

free parameters (1)
  • design parameters of the strictly convex function
    The specific strictly convex function used to encode the affine inequalities is chosen by the designer and may involve tunable weights or scaling factors.
axioms (1)
  • standard math A strictly convex function on a convex set possesses a unique minimizer
    Invoked to guarantee that the controller is well-defined and single-valued at every state.

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