Diffusion-Based Multiple-Shooting Indirect Optimal Control for Fuel-Optimal Spacecraft Trajectory Generation

Saeid Tafazzol; Ehsan Taheri; Ryne Beeson

eess.SYSep 12, 2026

Diffusion-Based Multiple-Shooting Indirect Optimal Control for Fuel-Optimal Spacecraft Trajectory Generation

Authors: Saeid Tafazzol, Ehsan Taheri, Ryne Beeson

Organizations: Department of Aerospace Engineering, Auburn University · Department of Mechanical and Aerospace Engineering, Princeton University

Abstract

Diffusion-based generative models (DMs) have found applications in control problems, and in particular robotics, where the DMs enable exploration of possible control solutions. A critical shortcoming of these applications is that they have lacked optimality guarantees. This is a problem for their potential use in fuel-optimal spacecraft trajectories that are characterized with long time-horizons and bang-bang profiles. Alternatively, indirect optimal control methods ensure explicit satisfaction of necessary conditions, but are highly sensitive to the initial costate estimation needed to solve the resulting Hamiltonian boundary-value problems (HBVPs). To alleviate this sensitivity and enlarge the convergence domain of HBVPs, advanced indirect methods have been developed that use smoothing approaches and continuation. We propose a diffusion-based multiple shooting indirect control method that combines the exploration capability of DMs with indirect method to generate fuel-optimal spacecraft trajectories. We benchmark our method against an advanced indirect method on a fuel-optimal Earth-Mars low-thrust transfer problem, showing higher convergence robustness than the advanced indirect method that is based on random costate initialization. Code and visualizations are available at https://saeidtafazzol.github.io/Diffusion_Indirect_Control/.

Figures & tables

Panel of Figure 1 — Fig. 1: transformer architecture used for the diffusion model.

Fig. 2: Dataset Coverage: To visualize the training dataset, the $14D$ effective input to the model for each trajectory was projected down to a 2D space using multidimensional scaling, which attempts to preserve the distances between trajectories in this 2D space.

Parameter	Value	Parameter	Value
LR decay epochs (linear)	24	Batch size	256
Forward diffusion levels $J$	5000	Epochs	30
Denoising steps (inference)	30	Weight decay	$10^{-6}$
Initial learning rate	$3\times 10^{-3}$	Dropout	0.1
Final learning rate	$10^{-5}$	Sequence length	32
IPOPT tolerance	$10^{-15}$	Embedding dimension	512

TABLE I: Model and inference parameters.

Shift	Ours	IPOPT	$m_{f}$	Shift	Ours	IPOPT	$m_{f}$
(d)	(%)	(%)	(kg)	(d)	(%)	(%)	(kg)
$-700$	$100.0$	$57.6$	$730.0$	$+50$	$100.0$	$56.8$	$659.6$
$-600$	$100.0$	$48.8$	$489.2$	$+100$	$100.0$	$52.4$	$722.7$
$-500$	$0.0$	$0.0$	—	$+200$	$0.0$	$0.0$	—
$-400$	$0.0$	$0.0$	—	$+300$	$0.0$	$0.0$	—
$-300$	$100.0$	$18.4$	$354.3$	$+400$	$0.4$	$6.0$	$285.0$

TABLE II: Convergence rates over 250 trials and optimal final mass per departure-date shift.

Fig. 3: Representative converged trajectories for transfer windows shifted by

-300

+100

, and

+500

days relative to the nominal departure epoch. AU denotes astronomical unit.

Fig. 4: Comparison of states, costates, switching function and throttle plots before and after IPOPT refinement. The diffusion model provides a near admissible solution and IPOPT ensures full admissibility.

Optimal Control

Optimal control studies how to choose actions over time to optimize performance while satisfying system dynamics and constraints.

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Transfer Learning of Multiobjective Indirect Low-Thrust Trajectories Using Diffusion Models and Markov Chain Monte Carlo

Jannik Graebner, Ryne Beeson

PhD Candidate, Department of Mechanical and Aerospace Engineering, Princeton University, NJ, USA. · Assistant Professor, Department of Mechanical and Aerospace Engineering, Princeton University, NJ, USA.

Markov Chain Monte Carlo Trajectory Optimization Diffusion Sampling Transfer Learning Multi-Objective Optimization

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Preliminary low-thrust spacecraft mission design is a global search problem characterized by a complex solution landscape, multiple objectives, and numerous local minima. During this phase, mission parameters are often not yet fully defined, requiring new solutions to be generated at a high cadence across varying parameter values. When combined with the indirect approach to optimal control, diffusion models can accelerate this search by learning distributions that represent high-quality initial costates. However, generating training data remains expensive, and opportunities exist to better exploit past data. We propose a transfer-learning framework that combines homotopy in a mission parameter with Markov chain Monte Carlo (MCMC) to generate training data more efficiently. The approach reformulates a multiobjective optimization problem as sampling from an unnormalized target distribution in costate space. We compare three MCMC algorithms on a planar multi-revolution transfer in the circular restricted three-body problem, with homotopy in the system mass parameter. The results show that gradient-based MCMC variants achieve the best trade-off between sample quality and computational cost. For the test transfer, the proposed framework generates 40 % more feasible solutions and achieves a higher-quality Pareto front than a state-of-the-art indirect approach based on adjoint control transformations and gradient-based optimization. Finally, the MCMC-generated samples are used to fine-tune a diffusion model conditioned on the mass parameter, enabling it to learn a global representation of the underlying solution distribution and efficiently generate new solutions. These findings establish the transfer-learning framework as a practical method for efficiently solving indirect trajectory optimization problems with varying parameters.

Abstract

Optimal Control

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