Reachability for Low-Thrust Trajectories via Maximum Initial Mass

Dario Izzo; Giacomo Acciarini; Zhong Zhang

arxiv: 2605.23770 · v3 · 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 2026-06-30 15:13 UTC · model grok-4.3

classification 📡 eess.SY astro-ph.EPcs.SYmath.OCphysics.space-ph

keywords reachability analysislow-thrust trajectoriesoptimal controlmaximum initial masssurrogate modelsneural networksspacecraft trajectory optimization

0 comments

The pith

Reachability of low-thrust targets reduces to computing 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.

The paper introduces a dual formulation of reachability analysis for low-thrust spacecraft trajectories. Instead of building reachable sets by solving many optimal control problems over grids of terminal states, it computes for each fixed target the largest initial mass (or sail-strength parameter) that still allows a successful transfer under given time and boundary conditions. A target counts as reachable exactly when the spacecraft's actual initial mass lies at or below this threshold value. The resulting scalar field over state space encodes the same feasibility information as a classical reachable set but arises from a single scalar optimization per target. Indirect optimal-control methods are derived for both electric low-thrust and solar-sail dynamics, and residual neural networks are trained to approximate the field rapidly for repeated queries.

Core claim

Determining the maximum initial mass for fixed transfer time and boundary conditions yields a scalar threshold whose comparison with the actual initial mass decides reachability, thereby converting the set-valued reachability question into a family of scalar optimization problems that can be solved by indirect methods and approximated by surrogate models.

What carries the argument

The maximum initial mass (MIM) obtained from the indirect optimal control formulation, which acts as the reachability threshold for each target.

If this is right

Reachability queries reduce to one scalar optimization per target rather than repeated forward simulations over a grid.
The resulting MIM field is smooth and supplies equivalent feasibility data to classical reachable sets.
Indirect formulations exist for both electric low-thrust and solar-sail dynamics.
Residual-network surrogates approximate the MIM field while preserving the numerical properties of the dual formulation.

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 be directly optimized over during mission-design loops to select initial mass or launch windows.
The same dual approach may apply to other continuous-thrust systems once their indirect optimality conditions are available.
Surrogate models trained on MIM values could support real-time feasibility checks inside onboard planners.

Load-bearing premise

The maximum initial mass computed from the indirect formulation exactly coincides with the boundary of reachability, so that a feasible trajectory exists if and only if the spacecraft starts with mass at or below this value.

What would settle it

Discovery of a feasible trajectory to a target when the initial mass exceeds the computed MIM value, or consistent failure to reach the target when the initial mass is below that value.

Figures

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

**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 statistics and subsequently clipped to a finite range, which improves conditioning and limits the influence of outliers. As shown in previous works, rotational invariant and Lambert solutions inputs are very … view at source ↗

**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.

Desk Editor's Note private letter to a colleague

The dual MIM reformulation is a clean idea that turns reachability into per-target scalar optimization, but indirect PMP solutions will not reliably mark the true boundary in these non-convex problems.

read the letter

The paper's core move is to flip the usual reachability question: instead of fixing initial mass and checking which targets are reachable, it solves for the largest initial mass that still allows a transfer to a fixed target under fixed time and thrust limits. That scalar value then acts as a reachability oracle. They derive the indirect necessary conditions for both low-thrust electric and solar-sail cases and train residual networks to approximate the resulting MIM field.

This is genuinely different from the standard grid-of-terminal-states approach and could cut computation for preliminary cislunar work. The surrogate section is straightforward but sensible; residual nets are a reasonable choice for the smoothness they want.

The soft spot is exactly the one the stress-test flags. Indirect methods satisfy Pontryagin conditions but return stationary points, not necessarily the global maximum initial mass. Low-thrust orbital problems are non-convex, so a local MIM can be lower than the true supremum. That breaks the claimed if-and-only-if equivalence to classical reachable sets. Nothing in the abstract or the described method indicates they run multiple starts, use global solvers, or provide numerical checks that the computed MIM actually matches the boundary obtained from direct methods.

The work is aimed at mission designers who need fast feasibility maps rather than rigorous set computation. It is coherent on its own terms and shows honest engagement with the optimal-control literature, so it is worth sending out for review. A referee can ask for the missing validation against known reachable sets and for evidence that local optima are not mis-classifying targets.

Referee Report

2 major / 1 minor

Summary. The paper claims that reachability for low-thrust and solar-sail trajectories can be assessed via a dual formulation that computes, for each fixed target and transfer time, the maximum initial mass (MIM) permitting a feasible transfer; a target is reachable precisely when the spacecraft initial mass is at or below this MIM value. This is obtained from indirect optimal-control formulations enforcing Pontryagin’s necessary conditions, yielding a smooth scalar field equivalent to classical reachable sets. The approach is further approximated by residual neural-network surrogates for rapid evaluation.

Significance. If the central equivalence holds, the reformulation would replace expensive gridding of reachable sets with per-target scalar optimizations and cheap surrogate queries, offering a practical tool for preliminary feasibility assessment in nonlinear regimes such as cislunar space. The explicit construction of indirect MIM problems and the reported preference for residual networks over fully connected nets constitute concrete, reproducible contributions that could be adopted by mission-design workflows.

major comments (2)

[Abstract] Abstract, dual-formulation paragraph: the claim that 'a target is reachable if the spacecraft's initial mass does not exceed this threshold' and that the MIM field 'encodes equivalent feasibility information to classical reachable sets' is load-bearing. Indirect methods solve the TPBVP arising from Pontryagin’s principle and therefore locate only stationary points; the low-thrust orbital problem is non-convex, so distinct control histories can produce distinct stationary values of initial mass. Without additional arguments or numerical evidence that the computed MIM is the global supremum, the claimed if-and-only-if equivalence to reachable sets does not follow.
[Abstract] Abstract, surrogate paragraph: the statement that the neural-network surrogates 'preserve the numerical advantages of the dual formulation' presupposes that the underlying MIM values are themselves globally optimal. If the indirect solutions are only locally optimal, the training data already embed classification errors; the surrogates then propagate rather than mitigate the mismatch with classical reachable sets.

minor comments (1)

[Abstract] The abstract supplies no quantitative validation (error metrics, comparison against grid-based reachable sets, or convergence statistics), making it impossible to assess how closely the computed MIM field matches classical results even on the tested cases.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments identifying the need for explicit support of global optimality in the indirect MIM formulation. We address each major comment below and will revise the manuscript accordingly to strengthen the equivalence claims.

read point-by-point responses

Referee: [Abstract] Abstract, dual-formulation paragraph: the claim that 'a target is reachable if the spacecraft's initial mass does not exceed this threshold' and that the MIM field 'encodes equivalent feasibility information to classical reachable sets' is load-bearing. Indirect methods solve the TPBVP arising from Pontryagin’s principle and therefore locate only stationary points; the low-thrust orbital problem is non-convex, so distinct control histories can produce distinct stationary values of initial mass. Without additional arguments or numerical evidence that the computed MIM is the global supremum, the claimed if-and-only-if equivalence to reachable sets does not follow.

Authors: We acknowledge that the non-convex nature of the problem means indirect methods yield stationary points that are not automatically guaranteed to be global maxima. In the revised manuscript we will add a dedicated subsection (in the MIM formulation section) that (i) states the conditions under which the computed stationary point is the global supremum for the low-thrust and solar-sail problems considered, and (ii) supplies numerical evidence obtained by cross-validating a representative set of cislunar and solar-sail test cases against direct collocation solutions. These additions will explicitly support the if-and-only-if equivalence asserted in the abstract. revision: yes
Referee: [Abstract] Abstract, surrogate paragraph: the statement that the neural-network surrogates 'preserve the numerical advantages of the dual formulation' presupposes that the underlying MIM values are themselves globally optimal. If the indirect solutions are only locally optimal, the training data already embed classification errors; the surrogates then propagate rather than mitigate the mismatch with classical reachable sets.

Authors: We agree that surrogate fidelity is contingent on the global optimality of the training labels. The new subsection and numerical cross-validation described in the response to the first comment will confirm that the MIM data used for training are globally optimal. With this verification in place, the claim that the surrogates preserve the dual formulation’s advantages will be justified; we will also edit the abstract to reference the added validation. revision: yes

Circularity Check

0 steps flagged

No circularity: dual MIM formulation is an independent reformulation

full rationale

The paper's central step is a reformulation of reachability as the computation of maximum initial mass (MIM) via indirect optimal control for fixed boundaries, asserting equivalence to classical reachable sets because a target is reachable iff m0 ≤ MIM. No equations, definitions, or claims in the abstract reduce this equivalence to a fitted parameter, self-definition, or load-bearing self-citation. The derivation relies on standard Pontryagin necessary conditions applied to the dual problem and is presented as self-contained against external benchmarks; the provided text contains none of the enumerated circular patterns. This is the expected honest non-finding for a reformulation paper.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; no free parameters, invented entities, or non-standard axioms are identifiable. The work assumes standard low-thrust and solar-sail dynamics without deriving them.

axioms (1)

domain assumption Standard models of electric low-thrust and solar-sail spacecraft dynamics
Invoked as the basis for the indirect MIM formulations without further justification in the abstract.

pith-pipeline@v0.9.1-grok · 5825 in / 1150 out tokens · 42118 ms · 2026-06-30T15:13:32.286514+00:00 · methodology

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discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

The Maximum Initial Mass
astro-ph.EP 2026-06 unverdicted novelty 7.0

Introduces maximum-initial-mass optimal control problem for low-thrust transfers, establishes correspondence to minimum-time extremals, and applies it to recover global solutions for a GTO-to-GEO benchmark.

Reference graph

Works this paper leans on

28 extracted references · 2 canonical work pages · cited by 1 Pith paper

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Reachable set computation for spacecraft relative motion with energy-limited low-thrust

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Izzo, D., M¨ artens, M., Beauregard, L., Bannach, M., Acciarini, G., Blazquez, E., Hadjiivanov, A., Grover, J., Heißel, G., Shimane, Y., et al. “Asteroid mining: ACT&Friends’ results for the GTOC12 problem.” Astrodynamics, Vol. 9, No. 1, pp. 19–40, 2025

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Bansal, S., Chen, M., Herbert, S., and Tomlin, C. J. “Hamilton-jacobi reachability: A brief overview and recent advances.” “2017 IEEE 56th annual con- ference on decision and control (CDC),” pp. 2242–

2017

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Fisac, J. F., Chen, M., Tomlin, C. J., and Sastry, S. S. “Reach-avoid problems with time-varying dy- namics, targets and constraints.” “Proceedings of the 18th international conference on hybrid systems: computation and control,” pp. 11–20. 2015

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Pretrained Approximators for Low- Thrust Trajectory Cost and Reachability

Zhang, Z., Acciarini, G., Izzo, D., Baoyin, H., and Topputo, F. “Pretrained Approximators for Low- Thrust Trajectory Cost and Reachability.” Journal of Guidance, Control, and Dynamics, 2026. To ap- pear

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P´ erez-Palau, D. and Epenoy, R. “Fuel optimization for low-thrust Earth–Moon transfer via indirect op- timal control.” Celestial Mechanics and Dynamical Astronomy, Vol. 130, No. 2, p. 21, 2018

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Indirect forward– backward shooting for low-thrust trajectory opti- mization in complex dynamics

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work page doi:10.2514/1.g007268 2023

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Practical tech- niques for low-thrust trajectory optimization with homotopic approach

Jiang, F., Baoyin, H., and Li, J. “Practical tech- niques for low-thrust trajectory optimization with homotopic approach.” Journal of guidance, control, and dynamics, Vol. 35, No. 1, pp. 245–258, 2012

2012

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New smoothing tech- niques for solving bang–bang optimal control prob- lems—numerical results and statistical interpreta- tion

Bertrand, R. and Epenoy, R. “New smoothing tech- niques for solving bang–bang optimal control prob- lems—numerical results and statistical interpreta- tion.” Optimal Control Applications and Methods, Vol. 23, No. 4, pp. 171–197, 2002

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2026

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Forward–Backward Indirect Shooting for Low- Thrust Trajectories with Interior Events

Zhang, Z., Yang, J., Izzo, D., and Topputo, F. “Forward–Backward Indirect Shooting for Low- Thrust Trajectories with Interior Events.” 30th In- ternational Symposium on Space Flight Dynamics, 2026

2026

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McInnes, C. R. Solar sailing: technology, dynamics and mission applications. Springer Science & Busi- ness Media, 2004

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