# Geometric Optimal Control and Applications to Aerospace

## TL;DR

This survey explores the integration of geometric optimal control and numerical continuation techniques to improve solutions for aerospace control problems, including attitude-trajectory control of launch vehicles.

## Contribution

It introduces a comprehensive approach combining geometric control theory with homotopy methods to enhance optimal control solutions in aerospace applications.

## Key findings

- Improved numerical methods for aerospace control problems.
- Analysis of chattering phenomena in control solutions.
- Application to launch vehicle attitude-trajectory optimization.

## Abstract

This survey article deals with applications of optimal control to aerospace problems with a focus on modern geometric optimal control tools and numerical continuation techniques. Geometric optimal control is a theory combining optimal control with various concepts of differential geometry. The ultimate objective is to derive optimal synthesis results for general classes of control systems. Continuation or homotopy methods consist in solving a series of parameterized problems, starting from a simple one to end up by continuous deformation with the initial problem. They help overcoming the difficult initialization issues of the shooting method. The combination of geometric control and homotopy methods improves the traditional techniques of optimal control theory. A nonacademic example of optimal attitude-trajectory control of (classical and airborne) launch vehicles, treated in details, illustrates how geometric optimal control can be used to analyze finely the structure of the extremals. This theoretical analysis helps building an efficient numerical solution procedure combining shooting methods and numerical continuation. Chattering is also analyzed and it is shown how to deal with this issue in practice.