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Mauro Pontani, Scuola di Ingegneria Aerospaziale, University of Rome “La Sapienza,” Rome, Italy,
Bruce A. Conway, Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, Urbana, IL
The determination of optimal (either minimum-time or minimum-propellant-consumption) space trajectories has been pursued for decades with different numerical optimization methods. In general, numerical optimization methods can be classified as deterministic or stochastic methods. Deterministic gradient-based methods assume the continuity and differentiability of the objective function to be minimized. In addition, gradient-based methods are local in nature and require the identification of a suitable first-attempt “solution” in the region of convergence, which is unknown a priori and strongly problem dependent. These circumstances have motivated the development of effective stochastic methods in the last decades. These algorithms are also referred to as evolutionary algorithms and are inspired by natural phenomena. Evolutionary computation techniques exploit a population of individuals, representing possible solutions to the problem of interest. The optimal solution is sought through cooperation and competition among individuals. The most popular class of these techniques is represented by the genetic algorithms (GA), which model the evolution of a species based on Darwin's principle of survival of the fittest. Differential evolution algorithms represent alternative stochastic approaches with some analogy with genetic algorithms, in the sense that new individuals are generated from old individuals and are eventually preserved after comparing them with their parents. Ant colony optimization is another method, inspired by the behavior of ants, whereas the simulated annealing algorithm mimics the equilibrium of large numbers of atoms during an annealing process.
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.
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].
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