On the Radau pseudospectral method: theoretical and implementation advances

In the last decades the theoretical development of more and more refined direct methods, together with a new generation of CPUs, led to a significant improvement of numerical approaches for solving optimal-control problems. One of the most promising class of methods is based on pseudospectral optimal control. These methods do not only provide an efficient algorithm to solve optimal-control problems, but also define a theoretical framework for linking the discrete numerical solution to the analytical one in virtue of the covector mapping theorem. However, several aspects in their implementation can be refined. In this framework SPARTAN, the first European tool based on flipped-Radau pseudospectral method, has been developed. This paper illustrates the aspects implemented for SPARTAN, which can potentially be valid for any other transcription. The novelties included in this work consist specifically of a new hybridization of the Jacobian matrix computation made of four distinct parts. These contributions include a new analytical formulation for expressing Lagrange cost function for open final-time problems, and the use of dual-number theory for ensuring exact differentiation. Moreover, a self-scaling strategy for primal and dual variables, which combines the projected-Jacobian rows normalization and the covector mapping, is described. Three concrete examples show the validity of the novelties introduced, and the quality of the results obtained with the proposed methods.