An Investigation of Optimal Powered Descent

Abstract

Achieving fuel-optimal pinpoint landings is a vital component of many missions. In this paper, the foundational components of optimal control theory known as Pontryagin’s Maximum Principle are derived, and an example is provided. The fuel-optimal trajectory for a one-dimensional lunar landing is then presented. The problem is then formulated in three-dimensions as a convex optimization problem. The main issue with this formulation is dealing with a non-convex constraint on the thrust, due to a non-zero lower bound. However, the constraint can be made to be convex through the use of a slack variable. Some results for a simulated landing on Mars are presented. Finally, a problem formulation using the so-called indirect method is shown. Principles of optimal control are applied, and a system of equations including the state variables and Hamiltonian is derived. Achieving convergence for the root finding algorithm is difficult due to sensitivities to the initial guess and numerical scaling.