Optimality conditions applied to free-time multi-burn optimal orbital transfers

TL;DR

This paper develops second-order optimality conditions for free-time multi-burn orbital transfers, ensuring candidate solutions are truly locally optimal, and demonstrates their application through numerical examples.

Contribution

It introduces second-order necessary and sufficient conditions for local optimality in multi-burn orbital transfers, enhancing the verification of candidate extremals.

Findings

Second-order conditions verify local optimality of transfer trajectories.

Numerical procedures effectively compute optimality conditions.

Two medium-thrust trajectories confirmed as locally optimal.

Abstract

While the Pontryagin Maximum Principle can be used to calculate candidate extremals for optimal orbital transfer problems, these candidates cannot be guaranteed to be at least locally optimal unless sufficient optimality conditions are satisfied. In this paper, through constructing a parameterized family of extremals around a reference extremal, some second-order necessary and sufficient conditions for the strong-local optimality of the free-time multi-burn fuel-optimal transfer are established under certain regularity assumptions. Moreover, the numerical procedure for computing these optimality conditions is presented. Finally, two medium-thrust fuel-optimal trajectories with different number of burn arcs for a typical orbital transfer problem are computed and the local optimality of the two computed trajectories are tested thanks to the second-order optimality conditions established in this paper.