A frequency-constrained geometric Pontryagin maximum principle on matrix Lie groups

TL;DR

This paper develops a geometric discrete-time Pontryagin maximum principle on matrix Lie groups that includes frequency constraints on controls, providing necessary conditions for optimality and demonstrating its application to spacecraft attitude control.

Contribution

It introduces a novel PMP framework on matrix Lie groups with frequency constraints, extending existing optimal control methods to more complex, constrained scenarios.

Findings

Derived first-order necessary conditions for optimality.

Formulated two-point boundary value problems solvable by shooting methods.

Validated the approach with a spacecraft attitude control example.

Abstract

In this article we present a geometric discrete-time Pontryagin maximum principle (PMP) on matrix Lie groups that incorporates frequency constraints on the controls in addition to pointwise constraints on the states and control actions directly at the stage of the problem formulation. This PMP gives first order necessary conditions for optimality, and leads to two-point boundary value problems that may be solved by shooting techniques to arrive at optimal trajectories. We validate our theoretical results with a numerical experiment on the attitude control of a spacecraft on the Lie group SO(3).