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arxiv: 2604.22244 · v1 · submitted 2026-04-24 · 💻 cs.RO

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Learning Control Policies to Provably Satisfy Hard Affine Constraints for Black-Box Hybrid Dynamical Systems

Aayushi Shrivastava , Kartik Nagpal , Sairam Jinkala , Jean-Baptiste Bouvier , Negar Mehr
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Pith reviewed 2026-05-08 11:33 UTC · model grok-4.3

classification 💻 cs.RO
keywords reinforcement learninghybrid dynamical systemssafety constraintsblack-box systemsaffine policiesreset mapsconstraint satisfaction
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The pith

RL policies made affine and repulsive near boundaries can provably keep black-box hybrid systems inside affine constraints without knowing their dynamics.

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

The paper establishes that reinforcement learning policies for black-box hybrid dynamical systems can be forced into an affine repulsive structure near affine constraint boundaries to guarantee that trajectories never enter the unsafe region, even when the nonlinear continuous dynamics are completely unknown. A second repulsive affine region is placed just before each reset map so that post-jump states also remain safe. Sufficient conditions are derived under which these policies satisfy the constraints in closed loop, and the resulting policies are shown to outperform reward-shaping and learned-CBF baselines on a constrained pendulum and a paddle juggler while never violating the limits.

Core claim

By restricting the learned policy to be affine and repulsive near the constraint boundaries for the unknown nonlinear dynamics and introducing a second repulsive affine region before each affine reset, the closed-loop trajectories of black-box hybrid systems provably satisfy the hard affine state constraints.

What carries the argument

An affine repulsive policy structure that outputs controls directing the state away from the unsafe set near each constraint boundary, plus a pre-reset repulsive zone that prevents post-jump violations.

Load-bearing premise

That an affine repulsive policy structure, without any knowledge of the unknown nonlinear dynamics, is sufficient to prevent constraint violations both during continuous flow and immediately after affine resets.

What would settle it

A concrete hybrid system, affine constraint, and learned policy satisfying the sufficient conditions for which a closed-loop trajectory starting from a safe initial state reaches the unsafe region either in flow or after a reset.

Figures

Figures reproduced from arXiv: 2604.22244 by Aayushi Shrivastava, Jean-Baptiste Bouvier, Kartik Nagpal, Negar Mehr, Sairam Jinkala.

Figure 1
Figure 1. Figure 1: Hybrid automaton representation of a two-mode hybrid system with view at source ↗
Figure 2
Figure 2. Figure 2: Phase portrait of the affine safety constraint view at source ↗
Figure 3
Figure 3. Figure 3: Actor-Critic Network Architecture (Modified Flow) view at source ↗
Figure 4
Figure 4. Figure 4: Constrained Pendulum System: (Left) Mode view at source ↗
Figure 5
Figure 5. Figure 5: Phase portrait of the constrained pendulum showing angle view at source ↗
Figure 6
Figure 6. Figure 6: Paddle Juggler System to illustrate the capability of our method. The network architecture and training pipeline are the same as in the constrained pendulum. The only difference is in training the affine policy for buffer BJ , where we try to satisfy the condition of Theorem (2) by randomly resetting the initial state within the buffer view at source ↗
Figure 7
Figure 7. Figure 7: Phase portrait of relative position s[0](t) and relative velocity s[1](t) for the paddle juggler under our policy. Red and Orange lines denote the constraints y(t) and yJ (t), respectively. Yellow and Brown regions denote the buffers B and BJ , respectively. (a) Trajectory starting inside B and (b) Trajectory starting inside BJ (a) Trajectories starting near y(t) and (b) Trajectories starting near yJ (t), … view at source ↗
read the original abstract

Ensuring safety for black-box hybrid dynamical systems presents significant challenges due to their instantaneous state jumps and unknown explicit nonlinear dynamics. Existing solutions for strict safety constraint satisfaction, like control barrier functions (CBFs) and reachability analysis, rely on direct knowledge of the dynamics. Similarly, safe reinforcement learning (RL) approaches often rely on known system dynamics or merely discourage safety violations through reward shaping. In this work, we want to learn RL policies which provably satisfy affine state constraints in closed loop for black-box hybrid dynamical systems with affine reset maps. Our key insight is forcing the RL policy to be affine and repulsive near the constraint boundaries for the unknown nonlinear dynamics of the system, providing guarantees that the trajectories will not violate the constraint. We further account for constraint violation due to instantaneous state jumps that occur due to impacts or reset maps in the hybrid system by introducing a second repulsive affine region before the reset that prevents post-reset states from violating the constraint. We derive sufficient conditions under which these policies satisfy safety constraints in closed loop. We also compare our approach with state-of-the-art reward shaping and learned-CBF methods on hybrid dynamical systems like the constrained pendulum and paddle juggler environments. In both scenarios, we show that our methodology learns higher quality policies while always satisfying the safety constraints.

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 learning RL policies for black-box hybrid dynamical systems (unknown nonlinear continuous dynamics, known affine resets) that provably satisfy hard affine state constraints. The key idea is to restrict policies to an affine repulsive structure near constraint boundaries (to repel trajectories from violation during flow) plus a second pre-reset repulsive affine region (to ensure post-jump states remain safe). Sufficient conditions are derived under which these structured policies guarantee closed-loop safety; experiments on constrained pendulum and paddle-juggler hybrids show the method learns higher-quality policies than reward-shaping or learned-CBF baselines while never violating constraints.

Significance. If the derived sufficient conditions truly guarantee safety for arbitrary unknown nonlinear f without hidden bounds or data-dependent certification of gains, the result would be significant: it supplies hard, model-free safety for hybrid systems via policy structure rather than CBFs or reachability that require known dynamics. The handling of affine resets via pre-reset repulsion is a concrete technical contribution, and the empirical outperformance on two hybrid benchmarks supports practicality. The approach could influence safe RL for impact-rich robotics if the guarantees hold.

major comments (2)
  1. [derivation of sufficient conditions] The derivation of sufficient conditions (abstract and § on policy structure) for the affine repulsive policy to prevent constraint violations under completely unknown nonlinear f(x, π(x)) must be examined for implicit dependence on bounds on ||f|| or its Lipschitz constant. For arbitrary black-box f, any finite repulsive gain can be overpowered near the boundary, so the conditions are load-bearing for the 'provably' claim; if they require a priori bounds or data-driven gain selection not stated in the abstract, the guarantee does not hold for general black-box hybrids.
  2. [hybrid reset handling] The pre-reset repulsive region is introduced to handle affine resets, but the interaction between the continuous repulsive policy, the reset map, and the unknown flow immediately before reset needs explicit verification that post-reset states remain inside the constraint for all possible pre-reset trajectories consistent with the unknown dynamics.
minor comments (2)
  1. [policy parameterization] Clarify the precise form of the affine repulsive policy (e.g., how the repulsive term is parameterized and whether it remains affine globally or only locally near boundaries) to aid reproducibility.
  2. [experiments] The experimental section should report the exact number of trials, variance in constraint satisfaction (even if zero), and whether any hyperparameter tuning was performed on the baselines for fair comparison.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed feedback. We address each major comment below, providing clarifications on the safety guarantees and indicating where revisions will strengthen the manuscript.

read point-by-point responses

  1. Referee: [derivation of sufficient conditions] The derivation of sufficient conditions (abstract and § on policy structure) for the affine repulsive policy to prevent constraint violations under completely unknown nonlinear f(x, π(x)) must be examined for implicit dependence on bounds on ||f|| or its Lipschitz constant. For arbitrary black-box f, any finite repulsive gain can be overpowered near the boundary, so the conditions are load-bearing for the 'provably' claim; if they require a priori bounds or data-driven gain selection not stated in the abstract, the guarantee does not hold for general black-box hybrids.

    Authors: The sufficient conditions in the policy structure section are derived to hold independently of any a priori bounds on ||f|| or its Lipschitz constant. By restricting the policy to an affine repulsive form in a neighborhood of each constraint boundary, the closed-loop dynamics are structurally forced to produce a strictly negative time derivative of the affine constraint function, repelling trajectories from violation for any continuous unknown nonlinear f. This follows directly from the affine parameterization without requiring knowledge of f or data-driven tuning of gains beyond the RL optimization itself. The abstract claim is therefore accurate as stated for general black-box hybrids. To address the referee's concern, we will add an explicit remark in the revised abstract and policy structure section confirming the absence of such bounds or selection procedures. revision: partial

  2. Referee: [hybrid reset handling] The pre-reset repulsive region is introduced to handle affine resets, but the interaction between the continuous repulsive policy, the reset map, and the unknown flow immediately before reset needs explicit verification that post-reset states remain inside the constraint for all possible pre-reset trajectories consistent with the unknown dynamics.

    Authors: We agree that an explicit verification of this interaction is valuable. Because the reset map is known and affine, the pre-reset repulsive region is sized so that its image under the reset lies strictly inside the safe set. The continuous repulsive policy ensures that trajectories remain in this region until a reset occurs. To cover all possible pre-reset flows under the unknown dynamics, we will add a supporting lemma in the hybrid reset handling section proving that the forward-invariant set induced by the repulsive policy before reset is mapped safely by the affine reset for any admissible pre-reset trajectory. This will be included in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper selects an affine repulsive policy structure by design and derives sufficient conditions for closed-loop constraint satisfaction under unknown nonlinear flow and known affine resets. No quoted step reduces the central claim to a fitted parameter renamed as a prediction, a self-definitional loop, or a load-bearing self-citation chain. The approach is an explicit ansatz plus forward derivation of inequalities, independent of the target safety result by construction. No patterns from the enumerated circularity kinds are exhibited.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

Abstract-only review prevents identification of specific fitted values or detailed axioms; the central claim rests on the policy being affine and repulsive plus affine resets.

axioms (2)
  • domain assumption Reset maps are affine
    Stated directly in the abstract as a property of the hybrid systems considered.
  • ad hoc to paper Affine repulsive policy structure yields closed-loop safety under derived sufficient conditions
    The paper's key insight and derivation rest on this structural assumption for unknown dynamics.
invented entities (1)
  • Repulsive affine region before reset no independent evidence
    purpose: Prevent post-reset constraint violations
    Introduced to handle instantaneous state jumps in hybrid systems.

pith-pipeline@v0.9.0 · 5549 in / 1351 out tokens · 84638 ms · 2026-05-08T11:33:00.179753+00:00 · methodology

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Reference graph

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