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arxiv: 2606.29289 · v1 · pith:IV7YTOK4new · submitted 2026-06-28 · 🌌 astro-ph.EP · astro-ph.IM

The Maximum Initial Mass

Dario Izzo , Giacomo Acciarini This is my paper

Pith reviewed 2026-06-30 02:41 UTC · model grok-4.3

classification 🌌 astro-ph.EP astro-ph.IM
keywords maximum initial masslow-thrust trajectoriesoptimal controlPontryagin maximum principleprimer vectorminimum-time problemcontinuation methodGTO-to-GEO transfer
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The pith

For fixed transfer time and final state, the maximum-initial-mass problem implies full-throttle control with primer-vector steering and a direct correspondence to minimum-time extremals.

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

The paper introduces the maximum-initial-mass problem as a standalone optimal-control formulation that, for fixed transfer time and final state, seeks the largest initial mass permitting a feasible low-thrust transfer. Its necessary conditions, obtained via Pontryagin's maximum principle, imply a full-throttle control law in the nondegenerate case together with the standard primer-vector steering direction. The work then proves a bidirectional correspondence: every minimum-time extremal induces a maximum-initial-mass extremal on its time interval, and conversely. This correspondence also recasts the terminal Hamiltonian condition as a gauge choice. The formulation supplies a smooth continuation method that recovers the global minimum-time solution for a benchmark GTO-to-GEO transfer while exposing additional extremal branches.

Core claim

The necessary conditions of the maximum-initial-mass problem imply a full-throttle control law in the nondegenerate case together with the standard primer-vector steering direction, and each minimum-time extremal induces a maximum-initial-mass extremal on the associated time interval and conversely.

What carries the argument

The maximum-initial-mass optimal-control problem, which maximizes initial spacecraft mass subject to fixed transfer time and fixed final state, with necessary conditions derived from Pontryagin's maximum principle.

If this is right

  • The maximum-initial-mass framework supplies a smooth continuation strategy for computing multiple-revolution low-thrust transfers.
  • Application to a GTO-to-GEO benchmark recovers the global minimum-time solution and identifies additional extremal branches.
  • Some trajectories previously reported in the literature are local maxima of transfer time along iso-M curves rather than local minima.

Where Pith is reading between the lines

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

  • Mass-continuation families could be used to locate global optima more reliably than direct time-minimization searches in high-revolution transfers.
  • The gauge interpretation of the terminal Hamiltonian may extend to other fixed-time problems in spacecraft trajectory optimization.
  • The correspondence suggests that minimum-time solutions sit inside one-parameter families indexed by initial mass.

Load-bearing premise

The analysis assumes the nondegenerate case and applies Pontryagin's maximum principle to the fixed-time, fixed-final-state problem without additional state or control constraints that would change the control structure.

What would settle it

An explicit nondegenerate extremal of the maximum-initial-mass problem that uses a non-full-throttle control law, or a minimum-time extremal that fails to generate a maximum-initial-mass extremal on the same interval.

read the original abstract

We introduce the maximum-initial-mass problem as a standalone optimal-control formulation for low-thrust trajectory design and analyze its structure within Pontryagin's framework. For fixed transfer time and final state, the formulation seeks the largest initial mass from which the transfer is feasible, and its necessary conditions imply a full-throttle control law in the nondegenerate case together with the standard primer-vector steering direction. We then establish a correspondence between extremals of the maximum-initial-mass and minimum-time problems, showing that each minimum-time extremal induces a maximum-initial-mass extremal on the associated time interval, and conversely. This viewpoint also clarifies the role of the terminal Hamiltonian condition in indirect formulations of minimum-time problems, which we interpret as a gauge choice rather than an independent necessary condition in the setting considered. Finally, we show that the maximum-initial-mass framework provides a smooth and effective continuation strategy for multiple-revolution low-thrust transfers. Applied to a benchmark GTO-to-GEO transfer, the approach recovers the global minimum-time solution, reveals additional extremal branches, and makes explicit that some trajectories previously reported in the literature correspond to local maxima of transfer time along iso-M curves rather than to local minima.

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

0 major / 2 minor

Summary. The manuscript introduces the maximum-initial-mass problem as a standalone optimal-control formulation for low-thrust trajectory design. For fixed transfer time and fixed final state, the problem seeks the largest feasible initial mass; its necessary conditions (via Pontryagin's maximum principle) imply a full-throttle control law together with the standard primer-vector steering direction in the nondegenerate case. The paper establishes a bijection between extremals of the maximum-initial-mass and minimum-time problems, interprets the terminal Hamiltonian condition as a gauge choice rather than an independent transversality condition, and presents the framework as a continuation strategy for multi-revolution transfers. An application to a GTO-to-GEO benchmark recovers the global minimum-time solution, identifies additional extremal branches, and reclassifies some previously reported trajectories as local maxima along iso-M curves.

Significance. If the stated bijection and control-structure results hold, the reformulation supplies a useful alternative viewpoint on indirect methods for low-thrust optimization and a practical continuation tool for multi-revolution problems. The explicit recovery of the known global solution on the benchmark and the identification of additional branches constitute concrete evidence of utility; the gauge interpretation of the terminal Hamiltonian also clarifies an aspect of existing minimum-time formulations.

minor comments (2)
  1. The abstract and introduction refer to 'the nondegenerate case' without an explicit definition or reference to the precise degeneracy condition used; a short clarifying sentence in §2 would help readers unfamiliar with the Pontryagin framework.
  2. The benchmark results are summarized qualitatively ('recovers the global minimum-time solution, reveals additional extremal branches'); a table or figure caption that lists the number of revolutions, final mass values, and transfer times for the recovered branches would strengthen the presentation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive review and the recommendation to accept the manuscript. The referee's summary correctly identifies the core contributions, including the bijection with minimum-time extremals and the utility of the maximum-initial-mass formulation as a continuation strategy.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation applies standard Pontryagin necessary conditions to the fixed-time, fixed-final-state, free-initial-mass problem. The full-throttle law and primer-vector direction follow directly from Hamiltonian maximization in the nondegenerate case; the extremal correspondence is obtained by costate rescaling and gauge reinterpretation of the terminal Hamiltonian value. These reductions are explicit consequences of the PMP axioms and the problem statement, with no fitted parameters renamed as predictions, no self-definitional loops, and no load-bearing self-citations. The nondegeneracy assumption is stated explicitly and the continuation strategy is presented as a practical consequence rather than a new theorem.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on the standard mathematical framework of optimal control theory; no free parameters, invented entities, or ad-hoc axioms are visible from the abstract.

axioms (1)
  • standard math Pontryagin's maximum principle supplies the necessary conditions for the fixed-time fixed-final-state problem
    Invoked to derive the full-throttle control law and primer-vector steering.

pith-pipeline@v0.9.1-grok · 5730 in / 1251 out tokens · 45615 ms · 2026-06-30T02:41:03.758233+00:00 · methodology

discussion (0)

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Reference graph

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