Hohmann Transfer via Constrained Optimization

TL;DR

This paper rigorously proves the global optimality of the classic Hohmann transfer using constrained optimization and calculus of variations, supported by numerical solutions verifying its optimality.

Contribution

It analytically confirms the global optimality of the Hohmann transfer and formulates it as boundary-value problems, providing new theoretical and numerical insights.

Findings

Hohmann transfer is globally optimal among two-impulse transfers.

Analytical proof of global optimality using Kuhn-Tucker conditions.

Numerical solutions via Matlab verify the transfer as a boundary-value problem solution.

Abstract

In the first part of this paper, inspired by the geometric method of Jean-Pierre Marec, we consider the two-impulse Hohmann transfer problem between two coplanar circular orbits as a constrained nonlinear programming problem. By using the Kuhn-Tucker theorem, we analytically prove the global optimality of the Hohmann transfer. Two sets of feasible solutions are found, one of which corresponding to the Hohmann transfer is the global minimum, and the other is a local minimum. In the second part, we formulate the Hohmann transfer problem as two-point and multi-point boundary-value problems by using the calculus of variations. With the help of the Matlab solver bvp4c, two numerical examples are solved successfully, which verifies that the Hohmann transfer is indeed the solution of these boundary-value problems. Via static and dynamic constrained optimization, the solution to the orbit transfer problem proposed by W. Hohmann ninety-two years ago and its global optimality are re-discovered.