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arxiv: 2605.23770 · v1 · pith:RCCCPGKLnew · submitted 2026-05-22 · 📡 eess.SY · astro-ph.EP· cs.SY· math.OC· physics.space-ph

Reachability for Low-Thrust Trajectories via Maximum Initial Mass

Giacomo Acciarini , Dario Izzo , Zhong Zhang This is my paper

Pith reviewed 2026-05-25 03:15 UTC · model grok-4.3

classification 📡 eess.SY astro-ph.EPcs.SYmath.OCphysics.space-ph
keywords reachability analysislow-thrust trajectoriesoptimal controlmaximum initial masssolar sailsneural network surrogatesfeasibility assessment
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The pith

Reachability of low-thrust spacecraft targets is determined by the maximum initial mass that permits a feasible transfer.

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

Classical methods build reachable sets by solving many optimal control problems on grids of terminal states. The paper reformulates the problem by fixing the target and computing the largest initial mass for which a transfer remains possible under time and thrust constraints. A target is reachable exactly when the spacecraft's starting mass stays below this computed threshold. This yields a smooth scalar field that carries the same feasibility information as the classical reachable set. The formulation applies to both electric low-thrust and solar-sail dynamics and supports construction of fast neural-network surrogates.

Core claim

For fixed transfer time and boundary conditions the maximum allowable initial mass (or sail-strength parameter) that permits a successful transfer serves as a reachability indicator: the target is reachable if and only if the actual initial mass does not exceed this value. The dual formulation converts reachability assessment into a scalar optimization problem whose solution produces an equivalent smooth feasibility map.

What carries the argument

The maximum-initial-mass (MIM) optimal-control formulation that turns the reachability question into the solution of a single scalar optimization problem per target.

If this is right

  • Indirect optimal-control methods become efficient reachability oracles for arbitrary targets.
  • Residual networks can approximate the MIM-based indicator with good accuracy and stability.
  • The scalar field enables rapid feasibility checks for preliminary mission design.
  • The approach remains valid for strongly nonlinear dynamics such as cislunar environments.

Where Pith is reading between the lines

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

  • The scalar MIM field could serve as a cost landscape for selecting among multiple candidate targets.
  • Surrogate models trained on MIM values might extend to varying transfer times without retraining the underlying optimal-control solver.
  • High-dimensional state spaces become more tractable because only one scalar needs to be optimized per query point rather than a full set boundary.

Load-bearing premise

Indirect optimal-control solvers can reliably locate the global maximum initial mass for arbitrary boundary conditions and nonlinear dynamics without being trapped by numerical instabilities or spurious local solutions.

What would settle it

For a fixed target, dynamics, and transfer time, compare the reachability classification given by the MIM threshold against an independent classical reachable-set computation performed on a fine grid around that target; mismatch would falsify the claimed equivalence.

Figures

Figures reproduced from arXiv: 2605.23770 by Dario Izzo, Giacomo Acciarini, Zhong Zhang.

Figure 1
Figure 1. Figure 1: Residual block • the normalised eccentricity of the departing orbit (ecc n); • the cosine and sine of the true anomaly at departure (cos f, sin f). All features are standardised based on the dataset statis￾tics and subsequently clipped to a finite range, which improves conditioning and limits the influence of out￾liers. As shown in previous works, rotational invariant and Lambert solutions inputs are very … view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of ground truth and surrogate reach [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
read the original abstract

Reachability analysis plays a central role in low-thrust spacecraft trajectory optimization by identifying which target states can be achieved under constraints on time, thrust, and propellant. Classical approaches construct reachable sets by solving many optimal control problems over grids of terminal states, requiring extensive forward simulations with fixed initial conditions. While effective, this approach is computationally expensive and becomes impractical for high-dimensional systems or strongly nonlinear dynamics, such as those encountered in cislunar environments or solar sail missions. This work introduces a dual formulation of the reachability problem. Instead of computing reachable sets directly, we determine, for fixed transfer time and boundary conditions, the maximum allowable initial mass (or, for solar sails, a scalar sail-strength parameter) that permits a successful transfer. A target is reachable if the spacecraft's initial mass does not exceed this threshold. This reformulation reduces reachability assessment to a scalar optimization problem for each target, producing a smooth scalar field that encodes equivalent feasibility information to classical reachable sets. We develop indirect maximum-initial-mass (MIM) formulations for both electric low-thrust and solar-sail dynamics and show how they can serve as efficient reachability oracles. Building on this formulation, we construct data-driven surrogate models to approximate the MIM-based reachability indicator. We investigate fully connected neural networks and demonstrate that residual networks provide the best trade-off between accuracy, training stability, and model complexity. The resulting surrogates enable rapid reachability evaluation while preserving the numerical advantages of the dual formulation, offering a practical tool for preliminary mission design and feasibility assessment.

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 / 1 minor

Summary. The paper claims that reachability for low-thrust trajectories can be reformulated as computing the maximum initial mass (MIM, or sail-strength parameter) permitting a successful transfer for fixed time and boundary conditions; a target is reachable if initial mass does not exceed this MIM threshold. This dual approach yields a smooth scalar field equivalent to classical reachable sets. Indirect optimal-control formulations are developed for electric and solar-sail dynamics, and residual neural networks are shown to provide the best surrogate approximation for rapid evaluation.

Significance. If the equivalence holds and the indirect method reliably recovers the global MIM, the reformulation offers a computationally lighter alternative to gridding reachable sets, particularly useful in high-dimensional nonlinear regimes such as cislunar or solar-sail problems. The surrogate construction, especially the reported advantage of residual networks for accuracy-stability trade-off, could enable practical preliminary mission design tools. No machine-checked proofs or reproducible code are mentioned.

major comments (2)
  1. [Abstract] The central claim that the MIM scalar field 'encodes equivalent feasibility information to classical reachable sets' is load-bearing yet unsupported by any numerical validation, error metrics, or direct comparison against classical reachable-set constructions. Without such evidence it is impossible to confirm that the indirect MIM formulation recovers the true supremum mass rather than a local value.
  2. [Abstract] The assumption that indirect methods for the MIM TPBVP reliably converge to the global optimum for arbitrary nonlinear boundary conditions (electric and solar-sail dynamics) is not accompanied by any discussion of costate initialization strategies, convergence safeguards, or failure modes. This directly affects the reliability of the reachability oracle.
minor comments (1)
  1. [Abstract] The abstract states that residual networks 'provide the best trade-off' but supplies no quantitative metrics (training loss, test error, or comparison tables) to support this ranking.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for these constructive comments highlighting the need for explicit validation of the MIM equivalence and for discussion of numerical robustness. Both points are well-taken and will be addressed through targeted additions to the manuscript.

read point-by-point responses

  1. Referee: [Abstract] The central claim that the MIM scalar field 'encodes equivalent feasibility information to classical reachable sets' is load-bearing yet unsupported by any numerical validation, error metrics, or direct comparison against classical reachable-set constructions. Without such evidence it is impossible to confirm that the indirect MIM formulation recovers the true supremum mass rather than a local value.

    Authors: We agree that the equivalence claim requires direct numerical substantiation beyond the theoretical reformulation. In the revised manuscript we will add a new subsection presenting side-by-side comparisons of MIM-derived reachability boundaries against classical gridded reachable sets for representative electric and solar-sail transfers. These comparisons will report quantitative metrics (Hausdorff distance between boundaries, classification accuracy on test grids, and success-rate statistics) and will include results from multiple random costate initializations to support that the reported MIM values correspond to the global supremum. revision: yes

  2. Referee: [Abstract] The assumption that indirect methods for the MIM TPBVP reliably converge to the global optimum for arbitrary nonlinear boundary conditions (electric and solar-sail dynamics) is not accompanied by any discussion of costate initialization strategies, convergence safeguards, or failure modes. This directly affects the reliability of the reachability oracle.

    Authors: The referee correctly notes the absence of this discussion. We will expand the indirect-method section to describe the costate initialization procedure (random sampling within physically motivated bounds combined with homotopy from simpler problems), the safeguards employed (multiple shooting attempts per target with different seeds and early termination on divergence), and observed failure modes together with their mitigation. Convergence statistics from the numerical experiments will be reported to quantify reliability. revision: yes

Circularity Check

0 steps flagged

No circularity: dual formulation is an independent mathematical reformulation

full rationale

The paper presents a dual reachability formulation equating reachability to initial mass below a computed MIM threshold for fixed time/boundaries. This is derived as a direct reformulation of the optimal control problem without any self-definition, fitted inputs renamed as predictions, or load-bearing self-citations that reduce the central claim to its own inputs. The indirect MIM method and subsequent surrogate training are presented as computational tools applied to the reformulated problem; no equations or steps reduce the claimed equivalence to a tautology or prior fitted result by construction. The derivation chain remains self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides insufficient detail to enumerate specific free parameters, axioms, or invented entities; none are explicitly introduced or fitted in the provided text.

pith-pipeline@v0.9.0 · 5825 in / 1141 out tokens · 26020 ms · 2026-05-25T03:15:53.547683+00:00 · methodology

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Reference graph

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