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arxiv: 2606.26352 · v1 · pith:PY2ABTFYnew · submitted 2026-06-24 · 📡 eess.SY · cs.SY· math.DS

Scalable Reachability Analysis of Linear Continuous Systems with Property-Driven Time-Step Adaptation

Mikkel Bj{\o}rn , Daniel H. Hansen , Grace Melchiors , Kim Guldstrand Larsen , Christian Schilling This is my paper

Pith reviewed 2026-06-26 01:09 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.DS
keywords reachability analysislinear time-invariant systemssafety verificationtime-step adaptationcontinuous-time systemsbounded inputsapproximation errorbenchmark evaluation
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The pith

Adapting time steps to the safety property lets linear system reachability use larger steps while still proving safety.

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

The paper presents a reachability algorithm for linear time-invariant systems with bounded inputs that chooses the largest time step for which safety can still be proved rather than using fixed steps or adapting only to approximation error. This property-driven choice, combined with optimizations that avoid repeated matrix exponential calculations and balance errors from states versus inputs, keeps the overall approximation moderate even at large steps. As a result the method requires substantially fewer discretization steps than prior approaches. A reader would care because continuous-time safety verification becomes practical on larger systems when the number of steps drops without losing soundness. The effectiveness is shown through consistent outperformance on the SLICOT benchmark suite.

Core claim

The algorithm selects the largest possible time step such that safety can still be proved for the given property. Optimizations prevent repeated expensive matrix exponential recomputation during discretization and maintain a careful balance between state and input approximation errors. These choices together produce only moderate over-approximation even when steps are enlarged, so far fewer steps suffice compared with fixed-step or error-driven methods, and the approach scales to large benchmark problems where it outperforms existing state-of-the-art reachability tools.

What carries the argument

Property-driven time-step adaptation that enlarges the step to the maximum value still allowing a safety proof.

If this is right

  • Reachability analysis finishes in fewer discretization steps for the same safety query.
  • Verification remains sound while using larger steps than error-only adaptation permits.
  • The same optimizations apply to other linear-system tools that rely on matrix exponentials.
  • Benchmark performance improves without changing the underlying continuous-time model.

Where Pith is reading between the lines

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

  • The same property-driven logic could be tested on systems whose inputs are not bounded but described by other constraints.
  • Integration with existing hybrid-system verifiers might reuse the step-selection routine without reimplementing discretization.
  • If the error-balance rule generalizes, similar adaptation could reduce step counts in related analyses such as stability checking.

Load-bearing premise

The chosen balance between state and input error bounds stays sound and does not hide unsafe behavior when time steps grow larger.

What would settle it

Run the algorithm on a linear system known to violate safety; if it reports safety or if the reported reachable set is smaller than the true reachable set on any SLICOT instance, the claim fails.

Figures

Figures reproduced from arXiv: 2606.26352 by Christian Schilling, Daniel H. Hansen, Grace Melchiors, Kim Guldstrand Larsen, Mikkel Bj{\o}rn.

Figure 1
Figure 1. Figure 1: Two overapproximations Rb [0,T](X0) of the reachable states. Each block corresponds to one step in time. The green sequence was obtained by a fixed￾step approach with a step size chosen such that safety (staying above the bottom unsafe region) is verified. The brown sequence was obtained by our approach, which chooses the time step depending on the distance to the unsafe region. Given a safety property in … view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of the discretization algorithms 1 and 2 for a system with [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of Algorithm 3 to LGG and BFFPSV. The step sizes for all [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of Algorithm 3 to LGG on the ISS model. The step sizes for [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of the discretization algorithms 1 and 2 on the Heat model. [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
read the original abstract

We study safety verification for linear time-invariant systems with bounded inputs in continuous time. The standard approach reduces to a reachability analysis in two steps: first discretize time and then apply a forward analysis in the discretized system. Existing algorithms use either a fixed time step or an adaptive time step that changes based on the approximation error compared to the underlying continuous system. In this paper, we present an efficient reachability algorithm that adapts the time step based on a given safety property. Essentially, our algorithm makes the largest possible time step such that it can still prove safety. For this approach to be scalable in practice, we discuss several optimizations such as avoiding the repeated expensive calculation of the matrix exponential during discretization and a careful balance how we tame the approximation error stemming from the states and the inputs. This allows our algorithm to yield a moderate approximation error even when using a large time step, thus requiring much fewer steps than prior algorithms. We demonstrate the effectiveness and scalability on the large-scale SLICOT benchmark suite, where our algorithm consistently outperforms other state-of-the-art approaches.

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 presents a reachability analysis algorithm for safety verification of linear time-invariant continuous-time systems with bounded inputs. It replaces fixed or error-driven time-step adaptation with a property-driven rule that selects the largest admissible time step still permitting a safety proof. The approach includes optimizations such as matrix-exponential caching and a tuned split between state and input approximation errors; the authors report that this yields moderate over-approximation error with far fewer steps than prior methods and demonstrate consistent outperformance on the SLICOT benchmark suite.

Significance. If the soundness of the property-driven adaptation rule is established, the method could materially improve the scalability of reachability-based verification for high-dimensional linear systems by reducing the number of discretization steps while preserving the ability to certify safety. The reported benchmark scaling and the explicit handling of input versus state error are practical strengths.

major comments (2)
  1. [§4] §4 (Property-Driven Time-Step Adaptation): the central claim that the algorithm selects the largest time step still allowing a safety proof rests on a 'careful balance' between state and input approximation errors. No derivation or inductive argument is supplied showing that this balance preserves the containment relation (reachable set of the continuous system contained in the computed over-approximation) when the step size is chosen from the property rather than from a uniform error bound; without such an argument the safety conclusion is not guaranteed.
  2. [§4.2] §4.2 (Error Propagation): the split of the discretization error into state and input components is described algorithmically but the recurrence that propagates these errors across variable-length steps is not stated explicitly. If any cross-term arising from the product of the state-transition matrix with the input-error set is omitted, the over-approximation may fail to contain the true reachable set precisely when the time step is enlarged.
minor comments (2)
  1. [Abstract, §1] The abstract and §1 repeatedly use the phrase 'moderate approximation error' without a quantitative definition; a short paragraph relating this phrase to the concrete Hausdorff-distance or support-function bounds used later would improve clarity.
  2. [Table 1] Table 1 (SLICOT results) reports run-times and step counts but does not list the final over-approximation diameters or the safety margins; adding these quantities would allow readers to judge whether the larger steps actually preserve useful tightness.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. The two major points both concern the rigor of the soundness argument for the property-driven step-size rule; we address them directly below and will strengthen the presentation accordingly.

read point-by-point responses

  1. Referee: [§4] §4 (Property-Driven Time-Step Adaptation): the central claim that the algorithm selects the largest time step still allowing a safety proof rests on a 'careful balance' between state and input approximation errors. No derivation or inductive argument is supplied showing that this balance preserves the containment relation (reachable set of the continuous system contained in the computed over-approximation) when the step size is chosen from the property rather than from a uniform error bound; without such an argument the safety conclusion is not guaranteed.

    Authors: We agree that an explicit inductive argument is required. The current manuscript only sketches the error-balancing condition; it does not contain a formal lemma proving that the containment relation is preserved when the step size is selected from the safety property. In the revision we will insert a new lemma (and its proof) in §4 that proceeds by induction over the sequence of (variable-length) steps, using the stated balance between state and input errors to show that the over-approximation remains valid at every step. revision: yes

  2. Referee: [§4.2] §4.2 (Error Propagation): the split of the discretization error into state and input components is described algorithmically but the recurrence that propagates these errors across variable-length steps is not stated explicitly. If any cross-term arising from the product of the state-transition matrix with the input-error set is omitted, the over-approximation may fail to contain the true reachable set precisely when the time step is enlarged.

    Authors: The referee is correct that the recurrence is only given algorithmically. The manuscript does not display the explicit recurrence relation that includes the cross-term arising from the state-transition matrix acting on the input-error set. We will add the full recurrence (with the cross-term made explicit) to §4.2 and verify that the over-approximation operator used in the implementation matches this recurrence. revision: yes

Circularity Check

0 steps flagged

No circularity: algorithmic method with independent optimizations

full rationale

The paper describes an algorithmic reachability procedure for linear systems that selects time steps based on a safety property rather than uniform error bounds. No equations, parameter fits, or self-citations appear in the abstract or description that would reduce any claimed result to its own inputs by construction. The central contribution consists of standard discretization steps plus practical optimizations (matrix-exponential caching and error balancing), all of which remain externally verifiable against benchmarks and do not rely on self-referential definitions or renamings.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are stated. The approach implicitly assumes standard properties of linear time-invariant systems and bounded inputs, but these are not enumerated.

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Reference graph

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