Abstract:
In this paper, two classic direct methods for numerical computation of optimal trajectories were revisited: the steepest descent method and the direct one based upon the second variation theory. The steepest descent method was developed for a Mayer problem of optimal control, with free final state and fixed terminal times. Terminal constraints on the state variables were considered through the penalty function method. The second method was based upon the theory of second variation and it involves the closed-loop solutions of a linear quadratic optimal control problem. The algorithm was developed for a Bolza problem of optimal control, with fixed terminal times and constrained initial and final states. Problems with free final time are also considered by using a transformation approach. An algorithm that combines the main characteristics of these methods was also presented. The methods were applied for solving two classic optimization problems - Brachistochrone and Zermelo - and their main advantages and disadvantages were discussed. Finally, the optimal space trajectories transference between coplanar circular orbits for different times of flight was calculated, using the proposed algorithm.
Keywords:
Optimization of trajectories; Numerical methods; Steepest descent method; Second-order gradient method