[Paper Review] Solving two-point boundary value problems using generating functions: Theory and Applications to optimal control and the study of Hamiltonian dynamical systems

Vincent Guibout, Daniel J. Scheeres|arXiv (Cornell University)|Oct 30, 2003

Adaptive Dynamic Programming Control15 references7 citations

TL;DR

This paper presents a generating function-based method to solve two-point boundary value problems in Hamiltonian systems by leveraging Hamilton-Jacobi theory and canonical transformations induced by phase flow. It enables efficient, direct solution of optimal control problems and phase space structure analysis—such as periodic orbits and equilibrium points—without initial guesses, with applications to spacecraft formation flight and trajectory design.

ABSTRACT

A methodology for solving two-point boundary value problems in phase space for Hamiltonian systems is presented. Using Hamilton-Jacobi theory in conjunction with the canonical transformation induced by the phase flow, we show that the generating functions for this transformation solve any two-point boundary value problem in phase space. Properties of the generating functions are exposed, we especially emphasize multiple solutions, singularities, relations with the state transition matrix and symmetries. Then, we show that using Hamilton's principal function we are also able to solve two-point boundary value problems, nevertheless both methodologies have fundamental differences that we explore. Finally, we present some applications of this theory. Using the generating functions for the phase flow canonical transformation we are able to solve the optimal control problem (without an initial guess), to study phase space structures in Hamiltonian dynamical systems (periodic orbits, equilibrium points) and classical targeting problems (this last topic finds its applications in the design of spacecraft formation trajectories, reconfiguration, formation keeping, etc...).

Motivation & Objective

• Address the inefficiency of traditional shooting methods in solving multiple two-point boundary value problems, especially in multi-spacecraft formation missions requiring n! trajectory solutions.
• Develop a systematic method to solve any two-point boundary value problem in phase space for n-degree-of-freedom Hamiltonian systems using generating functions derived from the Hamilton-Jacobi equation.
• Enable direct computation of trajectories, periodic orbits, and equilibrium points in Hamiltonian dynamical systems without iterative initial guess refinement.
• Extend the application of generating functions to optimal control problems, offering a non-iterative, analytical alternative to standard shooting-based approaches.
• Provide a theoretical and computational framework for solving complex astrodynamics problems such as Lambert’s problem and momentum-space boundary value problems.

Proposed method

• Use Hamilton-Jacobi theory to derive generating functions that represent the canonical transformation induced by the phase flow of a Hamiltonian system.
• Solve the Hamilton-Jacobi equation offline to obtain generating functions, which then map initial to final phase space states for any two-point boundary value problem.
• Apply Hamilton’s principal function as an alternative generating function, comparing its properties and limitations with the phase flow-based generating functions.
• Utilize the generating functions to compute trajectories by evaluating them at specified boundary conditions, bypassing numerical integration and iterative solvers.
• Leverage symmetries and singularities in the generating functions to analyze multiple solutions and structural features of phase space.
• Implement the method in practical applications such as optimal control and periodic orbit computation using numerical algorithms that converge locally in phase space.

Experimental results

Research questions

• RQ1 Can generating functions derived from the Hamilton-Jacobi equation be used to solve two-point boundary value problems in Hamiltonian systems without requiring initial guesses?
• RQ2 How do the solutions obtained via phase flow generating functions compare to those obtained via Hamilton’s principal function in terms of structure, multiple solutions, and singularities?
• RQ3 What are the implications of singularities and symmetries in the generating functions for the existence and multiplicity of solutions in boundary value problems?
• RQ4 To what extent can this method be applied to optimal control problems in aerospace systems, particularly in spacecraft formation flight?
• RQ5 Can this framework efficiently compute periodic orbits and equilibrium points in Hamiltonian systems such as the restricted three-body problem?

Key findings

• The generating functions for the phase flow canonical transformation solve any two-point boundary value problem in phase space by mapping initial to final states via solutions of the Hamilton-Jacobi equation.
• The method enables direct solution of optimal control problems without iterative refinement, as demonstrated in applications to spacecraft formation trajectories.
• Multiple solutions to boundary value problems are naturally captured through the structure of the generating functions, particularly when singularities or symmetries are present.
• The generating functions are related to the state transition matrix in linear systems, providing a geometric and analytical link between linearized dynamics and nonlinear solutions.
• Applications to the circular restricted three-body problem and Hill’s problem successfully computed periodic orbits and equilibrium points, with numerical validation via plots of solution sets.
• The method is computationally efficient for solving m boundary value problems at the cost of m function evaluations once generating functions are precomputed, making it scalable for multi-spacecraft missions.