Numerical and Analytical Methods for Global Optimization

Abstract

Global optimization is an important issue in the field of optimal aerospace trajectories. The joint use of global (approximate) numerical methods and of accurate local methods is one of the current approaches to global optimization. Nevertheless, the existence of global analysis techniques for investigating the possible multiplicity of results in optimization problems is of great interest. One of these techniques was the analysis of optimal trajectories through the Green’s theorem. This approach, formerly introduced by Miele, has been applied to find singular optimal solutions related to missile, spacecraft, and aircraft trajectories. Another approach is based on the Morse theory, which relates the number of singular points of the objective function to the topology of the space where this function is defined. In particular, the so-called Morse inequalities provide a lower bound on the number of the local minima of the objective function. In this paper, Miele’s and Morse’s approaches will be recalled and applied to some problems in flight mechanics.