Post-optimality Evaluation and Analysis of a Formation Flying Problem via a Gauss Pseudospectral Method

The post-optimality analysis of a tetrahedral formation flying optimal control problem is considered. In particular, this four-spacecraft orbit insertion problem is transcribed to a nonlinear programming problem (NLP) using a direct method called the Gauss pseudospectral method. The Karush-Kuhn-Tucker (KKT) conditions for this NLP are then derived and are compared to the conditions obtained via a Gauss pseudospectral discretization of the Hamiltonian boundary-value problem (HBVP) that arises from applying the calculus of variations. It is found that the optimal control obtained by solving the NLP is in excellent agreement with the control obtained from the Gauss pseudospectral discretization of the HBVP. The results obtained in this paper demonstrate the accuracy of the Gauss pseudospectral method and illustrate the usefulness of the Gauss pseu-dospectral method as a means of gaining insight into the structure of optimally controlled systems.