Universal Birkhoff Method for Computing Extremals in the Elliptic Restricted Three-Body Problem

Abstract

The computation of finite-thrust extremal arcs in the elliptic restricted three-body trajectory optimization problem is considered. Libration-point orbits are approximated to near-machine precision using a fast Fourier transform of the sampled values of the state vector at Chebyshev-Gauss-Lobatto points. Checkable optimality conditions are derived by applying Pontryagin's principle to minimum-time and time-constrained minimum-propellant problems. These necessary conditions include criteria for optimal departure and arrival points. For propellant consumption, a recently developed computational model is employed. This model is agnostic to the specific impulse of the propellant and varies as the inverse quadratic of a cosine term. Candidate optimal solutions are generated by combining the universal Birkhoff theory for trajectory optimization with the fast, guess-free spectral algorithm. The extremality of the Birkhoff-computed solution is validated against the Hamiltonian minimization condition and the transversality conditions. It is shown that the Birkhoff-theoretic spectral algorithm can generate verifiable extremals without any assistance or initialization from dynamical systems theory.