Geometry of optimal trajectories

Summary

Abstract

The optimization of orbital manoeuvres and lunar or interplanetary transfer paths is based on the use of numerical algorithms aimed at minimizing a specific cost functional. Despite their versatility, numerical algorithms usually generate results which are local in character. Geometrical methods can be used to drive the numerical algorithms towards the global optimal solution of the problems of interest. In the present paper, Morse inequalities and Conley's topological methods are applied in the context of some trajectory optimization problems.

Introduction

Geometrical methods and techniques of differential topology have been useful in the study of dynamical systems for a long time. Classical results are provided by Morse theory and in particular Morse inequalities. These relate the number of critical points of index k of a function f : M → R, defined on a manifold M, to the k-homology groups of M. The manifold M can be a finite dimensional manifold [Mor1], the infinite dimensional manifold of paths in a variational problem [Mor2], [PS] or the manifold of control functions in an optimal control problem [AV], [V2].

The gradient flow of a Morse function f defines a retracting deformation that maps M into neighborhoods of its critical points of index k. These neighborhoods are identified with cells of dimension k, then a cell decomposition of M is determined through the function f [Mil].