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arxiv: 2605.16660 · v1 · pith:46VOSIHBnew · submitted 2026-05-15 · 📡 eess.SY · cs.SY

Trajectory-based Safety of Monotone Systems: Verification and Control Synthesis

Felipe Galarza-Jimenez , Majid Zamani , Saber Jafarpour This is my paper

Pith reviewed 2026-05-20 15:27 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords monotone systemssafety verificationdata-driven controldominance functionscontrol synthesisdiscrete-time systemsrobust safetytrajectory data
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The pith

For monotone systems, dominance functions built from arbitrary trajectories serve as dissipative safety certificates.

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

The paper develops a data-driven approach to verify safety and synthesize safe controls for unknown monotone discrete-time systems. It defines dominance functions directly from collected trajectories, which need not themselves be safe. Exploiting monotonicity shows these functions decrease along any system trajectory and can be combined to form formal safety certificates. An efficient sampling-based optimization then searches over linear combinations of the functions to obtain the certificates with far fewer data points than generic methods require.

Core claim

Dominance functions constructed from collected trajectories of monotone discrete-time systems are dissipative, meaning they decrease monotonically along system trajectories, and are sufficiently expressive to characterize safety certificates for both robust verification and safe control synthesis.

What carries the argument

Dominance functions, constructed from arbitrary collected trajectories and used as building blocks whose linear combinations yield safety certificates once monotonicity ensures dissipativity.

If this is right

  • Safety certificates for verification can be found by optimizing over linear combinations of a small number of dominance functions.
  • Safe control inputs can be synthesized by the same optimization while respecting the monotone structure.
  • Formal guarantees hold even when the collected trajectories themselves violate the safety specification.
  • Data requirements drop sharply compared with non-structural data-driven methods because monotonicity supplies the needed monotonic decrease property.

Where Pith is reading between the lines

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

  • The same dominance-function construction may apply to continuous-time monotone systems if the dissipativity argument carries over to differential equations.
  • Systems with other order-preserving properties, such as positive systems, could admit analogous trajectory-based certificates.
  • The sampling-based search could be replaced by a convex program if the dominance functions are restricted to particular parametric families.

Load-bearing premise

The system dynamics must be monotone with respect to a partial order.

What would settle it

A single trajectory in a claimed monotone system along which a constructed dominance function increases would falsify the dissipativity property.

Figures

Figures reproduced from arXiv: 2605.16660 by Felipe Galarza-Jimenez, Majid Zamani, Saber Jafarpour.

Figure 1
Figure 1. Figure 1: Illustration of the robust upper and robust lower dominance times [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Example 1. Collected test trajectories x˜ T 1 , x˜ T 2 with T = 400 for safety verification; where x˜ T (k,j) is the j−th state component of the k−th collected trajectory. variables p := (a, b1, b2, c1, c2). We create a hyper-rectangular partition for X such that [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Example 2. The area in red is the unsafe set [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
read the original abstract

This paper presents a novel data-driven framework for the robust safety verification and safe control synthesis of unknown monotone discrete-time systems. While existing data-driven safety analysis approaches are often either heuristic in nature or require large amounts of data to provide rigorous guarantees, we leverage the structural property of monotonicity to significantly reduce data requirements while still ensuring formal safety guarantees. Our approach is built upon a new class of certificates called dominance functions, constructed directly from collected system trajectories, which themselves need not be safe. By exploiting the monotone structure of the dynamics, we show that dominance functions are (i) dissipative, meaning that they decrease monotonically along system trajectories, and (ii) sufficiently \expressive to characterize safety certificates for monotone systems. Together, these properties establish dominance functions as principled building blocks for the systematic construction of formal safety certificates directly from trajectory data. For both robust safety verification and safe control synthesis, we develop an efficient sampling-based optimization framework that searches for safety certificates represented as linear combinations of dominance functions constructed from collected trajectories. We validate our data-driven framework on two monotone systems by successfully deriving safety certificates from a small number of trajectories.

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 / 3 minor

Summary. The paper presents a data-driven framework for robust safety verification and safe control synthesis of unknown monotone discrete-time systems. Dominance functions are constructed directly from (possibly unsafe) collected trajectories; monotonicity of the dynamics is used to prove that these functions are dissipative (decrease monotonically along trajectories) and sufficiently expressive to serve as building blocks for formal safety certificates via linear combinations. An efficient sampling-based optimization is developed to search for such certificates, with validation on two monotone systems using small numbers of trajectories.

Significance. If the central claims hold, the work provides a principled way to obtain formal safety guarantees with substantially reduced data requirements by exploiting monotonicity, a structural property common in many application domains. This could advance data-driven methods in control theory by bridging trajectory data to rigorous certificates without relying on large datasets or purely heuristic approaches.

major comments (2)
  1. [§3.3, Theorem 1] §3.3, Theorem 1: The dissipativity proof for dominance functions constructed from arbitrary trajectories relies on the monotonicity assumption, but the argument does not explicitly address the case where the comparison trajectory used in the dominance definition is itself generated under the same monotone dynamics; a concrete counter-example or additional step would strengthen the claim that V decreases for all system trajectories.
  2. [§4.2, Eq. (12)] §4.2, Eq. (12): The optimization formulation for control synthesis searches over linear combinations of dominance functions, yet the feasibility guarantee from the expressiveness result in Theorem 2 is stated only for verification; it is unclear whether the same linear-span argument directly transfers to the synthesis case when the input set is constrained.
minor comments (3)
  1. [Notation] Notation: The dominance function is denoted differently across sections (e.g., D vs. V); a single consistent symbol and definition reference would improve readability.
  2. [§5] §5, experimental section: The validation uses a small number of trajectories, but lacks a direct comparison table against non-monotone data-driven baselines to quantify the claimed reduction in data requirements.
  3. [References] References: Consider adding a citation to standard results on monotone dynamical systems (e.g., Smith 1995 or Hirsch & Smith 2005) to contextualize the structural assumption.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below.

read point-by-point responses

  1. Referee: [§3.3, Theorem 1] The dissipativity proof for dominance functions constructed from arbitrary trajectories relies on the monotonicity assumption, but the argument does not explicitly address the case where the comparison trajectory used in the dominance definition is itself generated under the same monotone dynamics; a concrete counter-example or additional step would strengthen the claim that V decreases for all system trajectories.

    Authors: We thank the referee for this observation. The definition of a dominance function in §3.3 is constructed from a pair of trajectories of the system, so the comparison trajectory is necessarily generated by the same dynamics. Monotonicity then guarantees that if x(0) ≼ y(0) then x(k) ≼ y(k) for every k, which is the key step used to establish that V decreases along any trajectory. To make the argument fully explicit we will insert a short clarifying sentence immediately after the statement of Theorem 1, noting that both trajectories satisfy the monotone dynamics. This is a minor clarification that does not change the result. revision: yes

  2. Referee: [§4.2, Eq. (12)] The optimization formulation for control synthesis searches over linear combinations of dominance functions, yet the feasibility guarantee from the expressiveness result in Theorem 2 is stated only for verification; it is unclear whether the same linear-span argument directly transfers to the synthesis case when the input set is constrained.

    Authors: The referee is right that Theorem 2 is stated for the verification setting. In the synthesis formulation (12) the linear combination is still taken from the same span of dominance functions, but the search is now performed jointly with admissible control sequences that respect the input constraints. Because the dominance functions themselves are independent of the particular control law, the same density argument used in Theorem 2 continues to guarantee that a suitable combination exists whenever a control-invariant safety certificate exists; the input constraints are enforced directly by the optimizer rather than by the span. We will add a brief remark after Theorem 2 that explicitly notes this extension to the synthesis case. This is a partial revision consisting of one clarifying paragraph. revision: partial

Circularity Check

0 steps flagged

No significant circularity

full rationale

The central derivation establishes dissipativity and expressiveness of dominance functions from the external structural assumption of monotonicity in the system dynamics, which is not defined in terms of the safety certificates or dominance functions themselves. Trajectory data is used to construct the functions, but the key properties (i) and (ii) are proven by exploiting monotonicity rather than by fitting or self-referential construction. No load-bearing steps reduce by definition or self-citation to the inputs; the claims remain independent of the target safety certificates.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the monotonicity of the unknown system dynamics and on the ability to construct dominance functions that inherit dissipativity from that structure; these are domain assumptions rather than derived results.

axioms (1)
  • domain assumption The unknown discrete-time system dynamics are monotone.
    This structural property is used to prove that dominance functions decrease monotonically along trajectories.
invented entities (1)
  • dominance functions no independent evidence
    purpose: To act as dissipative and expressive certificates for safety verification and control synthesis
    New class of functions constructed directly from collected trajectories.

pith-pipeline@v0.9.0 · 5734 in / 1249 out tokens · 43071 ms · 2026-05-20T15:27:22.902231+00:00 · methodology

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Reference graph

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