Pseudospectral Methods for Infinite-Horizon Nonlinear Optimal Control Problems

A central computational issue in solving infinite-horizon nonlinear optimal control problems is the treatment of the horizon. In this paper, we directly address this issue by a domain transformation technique that maps the infinite horizon to a finite horizon. The transformed finite horizon serves as the computational domain for an application of pseudospectral methods. Although any pseudospectral method may be used, we focus on the Legendre pseudospectral method. It is shown that the proper class of Legendre pseudospectral methods to solve infinite-horizon problems are the Radau-based methods with weighted interpolants. This is in sharp contrast to the unweighted pseudospectral techniques for optimal control. The Legendre-Gauss-Radau pseudospectral method is thus developed to solve nonlinear constrained optimal control problems. An application of the covector mapping principle for the Legendre-Gauss-Radau pseudospectral method generates a covector mapping theorem that provides an efficient approach for the verification and validation of the extremality of the computed solution. Several example problems are solved to illustrate the ideas.