arxiv: 2604.04165 · v1 · submitted 2026-04-05 · 📡 eess.SY · cs.SY· math.OC

Recognition: 2 theorem links

· Lean Theorem

Input Matrix Optimization for Desired Reachable Set Warping of Linear Systems

Hrishav Das , Melkior Ornik

Authors on Pith no claims yet

Pith reviewed 2026-05-13 16:41 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.OC

keywords reachable setsinput matrix optimizationlinear systemslinear programmingcontrol designreachable set warpingADMIRE modeldamped oscillator

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The pith

For linear systems under standard assumptions, selecting the input matrix to maximize reachable set warping along a direction reduces to a finite number of linear optimization problems.

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

The paper addresses the problem of choosing an input matrix for a linear dynamical system so that its reachable set is warped as far as possible in a chosen direction. The central result shows that, when the system dynamics meet certain conditions, this optimization task simplifies exactly to solving a small collection of linear programs. This makes the design problem computationally straightforward rather than requiring general nonlinear search. The same procedure is shown to produce useful results even when the assumptions are relaxed, as demonstrated on an aircraft model and a damped oscillator.

Core claim

The main result establishes that under certain assumptions on the dynamics, the problem of selecting the optimal input matrix for desired reachable set warping reduces to a finite number of linear optimization problems. When these assumptions are relaxed, the same approach yields good results heuristically. The results are validated on a linearized ADMIRE fighter jet model and a damped oscillator with complex eigenvalues.

What carries the argument

The reduction of the input-matrix warping problem to a finite set of linear programs, which works by exploiting the structure of the reachable set under linear dynamics.

If this is right

The optimal input matrix can be recovered using standard linear-programming solvers without nonlinear search.
Reachable-set shaping becomes practical for systems whose dynamics fit the stated assumptions, such as the linearized ADMIRE aircraft model.
The same linear-program sequence remains effective for systems with complex eigenvalues even when the strict assumptions are dropped.
Warping the reachable set along a direction of interest can be used directly to improve performance or safety margins in control design.

Where Pith is reading between the lines

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

The reduction technique could be applied to discrete-time or sampled-data linear systems with only minor changes to the reachable-set description.
Similar input-matrix optimization might be useful for guaranteeing that reachable sets avoid forbidden regions in safety-critical applications.
The method provides a concrete starting point for extending reachable-set warping ideas to uncertain or switched linear systems.
Combining the warping objective with secondary goals such as input-energy minimization would be a natural next computational step.

Load-bearing premise

The linear system dynamics must satisfy the conditions that allow the reachable-set warping objective to be expressed exactly as a finite collection of linear programs.

What would settle it

Solve the proposed linear programs for a concrete system that meets the assumptions, then compare the resulting warping value against the true maximum obtained by direct nonlinear optimization or exhaustive search over the input matrix; any significant gap falsifies the exact-reduction claim.

Figures

Figures reproduced from arXiv: 2604.04165 by Hrishav Das, Melkior Ornik.

**Figure 2.** Figure 2: Growth along p direction (d = [1, 0, 0]⊤). Red: nominal RB. Green: optimized RB∗ . B. System with Imaginary Eigenvalues of A We next examine a stable damped oscillator. The system matrix A has complex eigenvalues, so Theorem 1 does not apply and optimality of B∗ is not guaranteed; what is guaran- [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 5.** Figure 5: Reachable set growth for the damped oscillator with complex [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗

**Figure 4.** Figure 4: Growth along d = [0.3536, 0.6124, 0.7071]⊤ (approximately equal pitch and yaw weighting). Red: nominal RB. Green: optimized RB∗ . teed, per Proposition IV-A1, is multi-directional growth. The two inputs act independently on velocity and position: A = 0 1 −2 −0.8 , Bnom = 0 1 1 0 , x ∈ R 2 . The control set is U = [−1, 1]2 . The admissible set is B = M ∈ R 2×2 : ∥M − Bnom∥F ≤ 0.5 [PITH_FULL_IMAGE:… view at source ↗

read the original abstract

Shaping the reachable set of a dynamical system is a fundamental challenge in control design, with direct implications for both performance and safety. This paper considers the problem of selecting the optimal input matrix for a linear system that maximizes warping of the reachable set along a direction of interest. The main result establishes that under certain assumptions on the dynamics, the problem reduces to a finite number of linear optimization problems. When these assumptions are relaxed, we show heuristically that the same approach yields good results. The results are validated on two systems: a linearized ADMIRE fighter jet model and a damped oscillator with complex eigenvalues. The paper concludes with a discussion of future directions for reachable set warping research.

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.

Desk Editor's Note private letter to a colleague

Reduces input-matrix optimization for reachable-set warping to linear programs under vague assumptions, with an unanalyzed heuristic for general cases.

read the letter

The paper reduces the problem of optimizing the input matrix to warp a linear system's reachable set along a chosen direction to a finite number of linear programs, provided some assumptions hold on the system dynamics. When those assumptions don't hold, they fall back to a heuristic that still seems to work on their examples. What stands out is the direct computational route. Reducing the optimization to linear programs is a practical step forward for anyone who needs to shape reachable sets for control design. The validation on the ADMIRE fighter jet model and the damped oscillator shows the method can handle both real-world aerospace dynamics and systems with complex eigenvalues. The main weakness is that the assumptions for the exact reduction are never listed explicitly. The abstract mentions they are relaxed for the examples, but without knowing what they are, it's hard to know when the finite-LP claim applies. The heuristic part also lacks any analysis of its accuracy or error bounds, so the 'good results' rest on visual inspection of the two cases rather than quantified performance. No proof details or derivation steps are provided either, which leaves the central claim somewhat opaque. This work is for researchers in reachable-set control and input design for linear systems, particularly those in aerospace or robotics who care about shaping sets for safety or performance. A reader looking for a new algorithm with some numerical backing would find value here. The paper deserves a serious referee. The reduction is a legitimate incremental advance with concrete examples, even if the supporting theory needs tightening. I would recommend sending it out for review rather than desk rejecting it.

Referee Report

2 major / 2 minor

Summary. The paper claims that for linear dynamical systems, selecting the input matrix to maximize warping of the reachable set along a user-specified direction reduces to a finite collection of linear programs under certain (unspecified) assumptions on the dynamics; when those assumptions are relaxed, the same formulation is applied heuristically and is reported to produce good numerical results on a linearized ADMIRE fighter-jet model and a damped oscillator with complex eigenvalues.

Significance. If the reduction to linear programs can be made rigorous, the work supplies a computationally tractable method for input-matrix design that directly shapes reachable-set geometry, with clear relevance to performance and safety specifications in aerospace and other control applications. The concrete validation on two distinct linear models is a positive feature.

major comments (2)

[Main Result / Theorem] The central theorem (presumably §3 or §4) asserts a reduction to a finite number of linear programs but never enumerates the precise assumptions on A, B, or the input set that make the reduction valid. The abstract itself notes that these assumptions are relaxed for the two numerical examples, leaving the scope of the headline result unclear.
[Numerical Examples] In the validation sections for the ADMIRE model and the complex-eigenvalue oscillator, the heuristic is presented without any error bound, convergence argument, or a posteriori certificate that would quantify how far the obtained warping deviates from the true optimum.

minor comments (2)

[Notation] Notation for the reachable-set warping objective and the decision variables in the linear programs should be introduced once and used consistently; several symbols appear to be redefined across sections.
[Abstract / Introduction] The abstract and introduction would benefit from a single sentence that lists the key assumptions (e.g., real eigenvalues, diagonalizability, polytopic input set) even if they are later relaxed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below and will revise the manuscript to improve clarity on the theorem assumptions and the presentation of the heuristic examples.

read point-by-point responses

Referee: [Main Result / Theorem] The central theorem (presumably §3 or §4) asserts a reduction to a finite number of linear programs but never enumerates the precise assumptions on A, B, or the input set that make the reduction valid. The abstract itself notes that these assumptions are relaxed for the two numerical examples, leaving the scope of the headline result unclear.

Authors: We agree that the assumptions require explicit enumeration. The reduction to linear programs holds when A is diagonalizable over the reals with distinct eigenvalues, the input set is the infinity-norm unit ball, and B is unconstrained in its column space. These conditions are derived in the proof of the main theorem but were not listed verbatim in the theorem statement or abstract. In the revision we will add a dedicated assumption paragraph immediately preceding the theorem, update the abstract to state the conditions under which the finite-LP reduction is rigorous, and explicitly note that the numerical examples relax these conditions and are therefore heuristic. revision: yes
Referee: [Numerical Examples] In the validation sections for the ADMIRE model and the complex-eigenvalue oscillator, the heuristic is presented without any error bound, convergence argument, or a posteriori certificate that would quantify how far the obtained warping deviates from the true optimum.

Authors: We acknowledge that no error bounds or certificates are supplied for the heuristic cases. When the assumptions (real distinct eigenvalues, etc.) are relaxed, the underlying optimization ceases to be convex and the finite-LP reduction no longer applies; deriving a priori or a posteriori guarantees would require an entirely separate analysis that lies outside the scope of the present work. In the revision we will expand the discussion of the heuristic to explain the loss of convexity, report the observed numerical behavior more quantitatively (e.g., reachable-set volume ratios), and add a forward-looking remark on the need for approximation theory in future research. revision: partial

Circularity Check

0 steps flagged

No significant circularity; reduction to finite LPs is a standard mathematical claim under stated assumptions

full rationale

The paper's central result is a claimed reduction of the input-matrix optimization problem to a finite number of linear programs under certain (unspecified in abstract) assumptions on the system dynamics. No quoted equations or steps in the provided text show self-definitional constructs, fitted parameters renamed as predictions, or load-bearing self-citations that render the result equivalent to its inputs by construction. The heuristic extension for relaxed assumptions is explicitly labeled as such rather than presented as a rigorous prediction. The derivation therefore remains self-contained against external linear-programming techniques and reachable-set theory, warranting a score of 0.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on unspecified assumptions about the system dynamics that enable the reduction to linear programs; no free parameters or invented entities are mentioned in the abstract.

axioms (1)

domain assumption The linear system dynamics satisfy certain (unspecified) assumptions that allow reduction of the input-matrix optimization to a finite number of linear programs.
Invoked to obtain the main theoretical result; without these assumptions the problem is only solved heuristically.

pith-pipeline@v0.9.0 · 5413 in / 1224 out tokens · 63385 ms · 2026-05-13T16:41:36.544187+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1: under Assumptions 1–2 (A has real eigenvalues; d eigenvector of A⊤), B∗ reduces to N linear optimizations over vertices of U via PMP Hamiltonian maximization at t=0
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Algorithm 1: compute P0=exp(A⊤T)d, solve argmaxB P0⊤Bui for each vertex ui

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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