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arxiv: 2604.08327 · v1 · submitted 2026-04-09 · 🧮 math.OC · cs.SY· eess.SY

Finite-time Reachability for Constrained, Partially Uncontrolled Nonlinear Systems

Ram Padmanabhan , Melkior Ornik This is my paper

Pith reviewed 2026-05-10 17:14 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY
keywords nonlinear systemsfinite-time reachabilitypartial control lossconstrained controllinear approximationtime partitioningdriftless systems
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The pith

Nonlinear systems with partial control loss can reach target states in finite time using linear approximations and time partitioning.

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

The paper develops a control technique for constrained nonlinear systems that have lost authority over some inputs. It approximates the dynamics with a simple linear driftless system at the starting point and divides the total time into smaller and smaller intervals. In each interval, inputs are chosen based on the approximation to steer towards the target. If the total time is bounded, the error stays limited in the real system, and it goes to zero with finer partitions even with the uncontrolled inputs acting.

Core claim

The technique constructs a partition of the finite time horizon into successively smaller intervals and designs controlled inputs based on the linear driftless approximation in each partition. Under conditions bounding the time horizon length, these inputs yield bounded error from the target in the nonlinear system, with the error reducing to zero as partitions shorten despite uncontrolled inputs.

What carries the argument

Successive partitioning of the time horizon combined with inputs from the linear driftless approximation at the initial state, which keeps error growth controlled in the original nonlinear dynamics.

If this is right

  • The target state becomes reachable in finite time despite the partial loss of control authority.
  • The state error remains bounded whenever the total time horizon satisfies the given length conditions.
  • Refining the time partitions drives the error from the target arbitrarily close to zero.
  • The sequence of inputs works on the fighter jet model even after losing authority over one input.

Where Pith is reading between the lines

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

  • The same partitioning idea could be combined with other local approximations to handle different classes of uncertainties or disturbances.
  • The result indicates that sufficiently fine time discretization can compensate for model mismatch or external effects in reachability problems.
  • Hardware experiments on vehicles or robots experiencing actuator failure would provide a direct test of whether the error reduction holds in practice.

Load-bearing premise

A linear driftless approximation exists at the initial state and the time horizon length can be bounded so that approximation errors remain manageable.

What would settle it

Run the fighter jet simulation or a similar nonlinear model with a fixed time horizon while making partitions successively finer and check whether the final state error approaches zero; persistent nonzero error when the initial approximation is inaccurate would disprove the claim.

Figures

Figures reproduced from arXiv: 2604.08327 by Melkior Ornik, Ram Padmanabhan.

Figure 1
Figure 1. Figure 1: Partitioning the interval [0, 𝑡 𝑓 ] using the sequence {𝑡𝑛}. time despite such a loss in control authority. This problem is formally stated below. Problem 1. Given an initial state 𝑥0 ∈ R 𝑑 , a target state 𝑥𝑡𝑔 ∈ R 𝑑 a fixed final time 𝑡 𝑓 and a prescribed radius 𝜀 > 0, design the controlled input 𝑢 𝑐 ∈ Uc such that k𝑥𝑡𝑔 − 𝑥(𝑡 𝑓 ) k∞ ≤ 𝜀 despite the effect of 𝑢 𝑢𝑐 ∈ Uuc . Define 𝑔0 := 𝑔(𝑥0), 𝑔 𝑐 0 := 𝑔 𝑐 (… view at source ↗
Figure 2
Figure 2. Figure 2: The function ℎ(𝑡) and its behavior based on 𝑑ℎ(𝑡) 𝑑𝑡 . Then, the relation 𝑒𝑛+1 ≤ 1 2 𝑒𝑛 holds, and lim𝑛→∞ 𝑒𝑛 = 0. Given some 𝜀 > 0, there thus exists some 𝑛1 such that 𝑒𝑛1 ≤ 𝜀. Proof. Recall the properties Δ𝑡𝑛+1 = 1 2 Δ𝑡𝑛, lim𝑛→∞ Δ𝑡𝑛 = 0. Using a Taylor series expansion, we know 𝑒 Δ𝑡𝑛+1𝐷𝑆 − 1 = Õ∞ 𝑘=1 (Δ𝑡𝑛+1𝐷𝑆) 𝑘 𝑘! = Õ∞ 𝑘=1  1 2  𝑘 (Δ𝑡𝑛𝐷𝑆) 𝑘 𝑘! ≤ 1 2 Õ∞ 𝑘=1 (Δ𝑡𝑛𝐷𝑆) 𝑘 𝑘! = 1 2  𝑒 Δ𝑡𝑛𝐷𝑆 − 1  , since  1… view at source ↗
Figure 3
Figure 3. Figure 3: Illustrating the convergence behavior based on The [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: State trajectories and control inputs for the ADMIRE fighter-jet model, reaching the target state [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
read the original abstract

This paper presents a technique to drive the state of a constrained nonlinear system to a specified target state in finite time, when the system suffers a partial loss in control authority. Our technique builds on a recent method to control constrained nonlinear systems by building a simple, linear driftless approximation at the initial state. We construct a partition of the finite time horizon into successively smaller intervals, and design controlled inputs based on the approximate dynamics in each partition. Under conditions that bound the length of the time horizon, we prove that these inputs result in bounded error from the target state in the original nonlinear system. As successive partitions of the time horizon become shorter, the error reduces to zero despite the effect of uncontrolled inputs. A simulation example on the model of a fighter jet demonstrates that the designed sequence of controlled inputs achieves the target state despite the system suffering a loss of control authority over one of its inputs.

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

1 major / 2 minor

Summary. This paper introduces a partitioning-based control strategy for achieving finite-time reachability in constrained nonlinear systems subject to partial loss of control authority. Building on a linear driftless approximation at the initial state, the method designs controlled inputs over successively refined time partitions. The authors prove that, under conditions bounding the time horizon, the state error remains bounded and converges to zero as the maximum partition length tends to zero, despite the presence of uncontrolled inputs. The approach is validated through a simulation on a fighter jet model.

Significance. If the central claim holds, this work extends approximation-based finite-time control methods to systems with actuator failures or persistent disturbances, which is relevant for applications in aerospace and robotics. The simulation on the fighter jet model provides concrete evidence of practical applicability in a high-dimensional nonlinear setting with constraints.

major comments (1)
  1. Abstract and main theorem: the claim that the error to the target tends to zero as maximum partition length δt → 0 despite uncontrolled inputs v(·) is load-bearing but appears at risk. The linear driftless approximation is constructed only at the initial state and the same approximate dynamics are used across all partitions, so the controlled input sequence u(·) cannot depend on realized values of v(·). Each interval therefore contributes an integrated displacement of order δt from the uncontrolled vector field; summing over T/δt intervals produces an O(T) term independent of δt. The proof must explicitly show this term is cancelled by the controlled inputs or absorbed into a uniformly vanishing higher-order remainder; otherwise the limit cannot hold for nonzero bounded v.
minor comments (2)
  1. The assumptions on the uncontrolled inputs (boundedness, regularity) should be stated explicitly and early, as they are central to the error analysis.
  2. In the simulation example, report quantitative error values for successively smaller partition lengths and specify the uncontrolled input trajectory used, to allow direct comparison with the theoretical convergence claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying a key point that requires clarification in the proof of the main convergence result. We address the concern below and will revise the manuscript to strengthen the exposition.

read point-by-point responses

  1. Referee: Abstract and main theorem: the claim that the error to the target tends to zero as maximum partition length δt → 0 despite uncontrolled inputs v(·) is load-bearing but appears at risk. The linear driftless approximation is constructed only at the initial state and the same approximate dynamics are used across all partitions, so the controlled input sequence u(·) cannot depend on realized values of v(·). Each interval therefore contributes an integrated displacement of order δt from the uncontrolled vector field; summing over T/δt intervals produces an O(T) term independent of δt. The proof must explicitly show this term is cancelled by the controlled inputs or absorbed into a uniformly vanishing higher-order remainder; otherwise the limit cannot hold for nonzero bounded v.

    Authors: We appreciate the referee drawing attention to the need for an explicit accounting of the uncontrolled inputs in the error analysis. The proof of the main theorem proceeds by comparing the true nonlinear trajectory (under the designed piecewise-constant u and arbitrary bounded v) to the trajectory of the fixed linear driftless approximation. Because the partition length δt is small, each interval contributes a local approximation error of order O(δt²) from the Taylor remainder of the vector field (including the term involving v), by the C¹ regularity and boundedness assumptions. Summing these local errors over T/δt intervals and applying a discrete Gronwall inequality yields a global state error of order O(δt) that vanishes uniformly as δt → 0; the integrated contribution of v is absorbed into this higher-order remainder rather than producing an independent O(T) term, because the control sequence u is chosen to steer the approximate system exactly to the target and the local Lipschitz constants remain controlled over the fixed, bounded horizon T. We will add an auxiliary lemma in the revised proof that isolates the uncontrolled displacement and shows it is dominated by the vanishing approximation error. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation relies on independent approximation and partitioning arguments

full rationale

The paper's central result is a proof that controlled inputs designed from a linear driftless approximation at the initial state, when applied over successively finer partitions of a bounded time horizon, drive the nonlinear system to the target with error vanishing as maximum partition length tends to zero. This construction and error analysis are self-contained and do not reduce any claimed prediction or uniqueness statement to a fitted parameter, self-citation, or definitional renaming. The reference to a prior method supplies context but is not invoked as a load-bearing uniqueness theorem or ansatz that forces the result; the error bound is derived directly from the system dynamics and partition properties.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper depends on standard assumptions from nonlinear control theory and a previously published approximation method.

axioms (2)
  • domain assumption System dynamics allow for a linear driftless approximation at the initial state
    This is the foundation for designing inputs in each partition.
  • domain assumption The time horizon can be bounded to ensure error convergence
    Required for the proof that error goes to zero.

pith-pipeline@v0.9.0 · 5455 in / 1252 out tokens · 40202 ms · 2026-05-10T17:14:06.189678+00:00 · methodology

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Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

  1. [1]

    H. K. Khalil, Nonlinear Systems , 3rd ed. Upper Saddle River, NJ, USA: Prentice Hall, 2002

    work page 2002

  2. [2]

    V . I. Utkin, Sliding Modes in Control and Optimization , ser. Commu- nications and Control Engineering. Heidelberg, Germany: Springer- Verlag, 1992

    work page 1992

  3. [3]

    Control of nonlinear systems using terminal sliding modes,

    S. T. Venkataraman and S. Gulati, “Control of nonlinear systems using terminal sliding modes,” in 1992 American Control Conference , Chicago, IL, USA, June 1992, pp. 891–893

    work page 1992

  4. [4]

    Grüne and J

    L. Grüne and J. Pannek, Nonlinear Model Predictive Control: Theory and Algorithms , 2nd ed., ser. Communcations and Control Engineer- ing. Cham, Switzerland: Springer Cham, 2018

    work page 2018

  5. [5]

    Russia’s Nauka module briefly tilts space station with unplanned thruster fire,

    M. Bartels, “Russia’s Nauka module briefly tilts space station with unplanned thruster fire,” Aug. 2021. [Online]. Available: https://www.space.com/nauka-module-thruster-fire-tilts-space-station

    work page 2021

  6. [6]

    K. Zhou, J. C. Doyle, and K. Glover, Robust and Optimal Control . Hoboken, NJ, USA: Prentice-Hall, 1996

    work page 1996

  7. [7]

    Zhou and J

    K. Zhou and J. C. Doyle, Essentials of Robust Control . Hoboken, NJ, USA: Prentice-Hall, 1997

    work page 1997

  8. [8]

    Qu, Robust control of nonlinear uncertain systems

    Z. Qu, Robust control of nonlinear uncertain systems . Hoboken, NJ, USA: John Wiley & Sons, Inc., 1998

    work page 1998

  9. [9]

    Robust control of nonlinear systems with parametric uncertainty,

    Q. Wang and R. F. Stengel, “Robust control of nonlinear systems with parametric uncertainty,” Automatica, vol. 38, no. 9, pp. 1591–1599, Sept. 2002

    work page 2002

  10. [10]

    Robust control of nonlinear sys- tems: Compensating for uncertainty,

    F. Lin, R. D. Brandt, and J. Sun, “Robust control of nonlinear sys- tems: Compensating for uncertainty,” International Journal of Control , vol. 56, no. 6, pp. 1453–1459, 1992

    work page 1992

  11. [11]

    Robust control of a class of uncertain nonlinear systems,

    Y . Wang, L. Xie, and C. E. De Souza, “Robust control of a class of uncertain nonlinear systems,” Systems & control letters , vol. 19, no. 2, pp. 139–149, Aug. 1992

    work page 1992

  12. [12]

    Time-optimal control of a multidimensional integrator chain with applications,

    M. Leomanni, G. Costante, and F. Ferrante, “Time-optimal control of a multidimensional integrator chain with applications,” IEEE Control Systems Letters, vol. 6, pp. 2371–2376, Feb. 2022

    work page 2022

  13. [13]

    Liberzon, Calculus of Variations and Optimal Control Theory: A Concise Introduction

    D. Liberzon, Calculus of Variations and Optimal Control Theory: A Concise Introduction . Princeton, NJ, USA: Princeton University Press, 2012

    work page 2012

  14. [14]

    Quantitative resilience of linear driftless systems,

    J.-B. Bouvier, K. Xu, and M. Ornik, “Quantitative resilience of linear driftless systems,” in SIAM Conference on Control and its Applications, Spokane, W A, USA, July 2021, pp. 32–39

    work page 2021

  15. [15]

    Resilience of linear systems to partial loss of control authority,

    J.-B. Bouvier and M. Ornik, “Resilience of linear systems to partial loss of control authority,” Automatica, vol. 152, June 2023

    work page 2023

  16. [16]

    Energetic resilience of linear driftless systems,

    R. Padmanabhan and M. Ornik, “Energetic resilience of linear driftless systems,” in 11th IFAC Symposium on Robust Control Design , Porto, Portugal, July 2025

    work page 2025

  17. [17]

    Approximate energetic resilience of nonlinear systems under partial loss of control authority,

    ——, “Approximate energetic resilience of nonlinear systems under partial loss of control authority,” Automatica, vol. 187, May 2026

    work page 2026

  18. [18]

    Designing resilient linear systems,

    J.-B. Bouvier and M. Ornik, “Designing resilient linear systems,” IEEE Transactions on Automatic Control, vol. 67, no. 9, pp. 4832–4837, Sep. 2022

    work page 2022

  19. [19]

    Ignore drift, embrace simplicity: Constrained nonlinear control through driftless approximation,

    R. Padmanabhan and M. Ornik, “Ignore drift, embrace simplicity: Constrained nonlinear control through driftless approximation,” 2025. [Online]. Available: https://arxiv.org/abs/2509.06188

    work page arXiv 2025

  20. [20]

    C. P . Niculescu and L.-E. Persson, Convex Functions and Their Applications: A Contemporary Approach . New Y ork, NY, USA: Springer, 2006

    work page 2006

  21. [21]

    T. M. Apostol, Calculus: Volume 1 , 2nd ed. Hoboken, NJ, USA: John Wiley & Sons, Inc., 1967

    work page 1967

  22. [22]

    ADMIRE The aero-data model in a research environment version 4.0, model description,

    L. Forssell and U. Nilsson, “ADMIRE The aero-data model in a research environment version 4.0, model description,” FOI – Swedish Defence Research Agency, Tech. Rep. FOI-R–1624–SE, Dec. 2005. [Online]. Available: https://www.foi.se/rest-api/report/ FOI-R--1624--SE

    work page 2005

  23. [23]

    UA V trajectory tracking under wind disturbance based on novel antidisturbance sliding mode control,

    Q. Wang, W. Wang, and S. Suzuki, “UA V trajectory tracking under wind disturbance based on novel antidisturbance sliding mode control,” Aerospace Science and Technology , vol. 149, June 2024

    work page 2024

  24. [24]

    An adaptive actuator failure com- pensation controller using output feedback,

    G. Tao, S. Chen, and S. Joshi, “An adaptive actuator failure com- pensation controller using output feedback,” IEEE Transactions on Automatic Control , vol. 47, no. 3, pp. 506–511, Mar. 2002

    work page 2002

  25. [25]

    A review of fault tolerant control systems: Advancements and applications,

    A. A. Amin and K. M. Hasan, “A review of fault tolerant control systems: Advancements and applications,” Measurement, vol. 143, pp. 58–68, Sept. 2019

    work page 2019