Connections between the covector mapping theorem and convergence of pseudospectral methods for optimal control

In recent years, many practical nonlinear optimal control problems have been solved by pseudospectral (PS) methods. In particular, the Legendre PS method offers a Covector Mapping Theorem that blurs the distinction between traditional direct and indirect methods for optimal control. In an effort to better understand the PS approach for solving control problems, we present consistency results for nonlinear optimal control problems with mixed state and control constraints. A set of sufficient conditions is proved under which a solution of the discretized optimal control problem converges to the continuous solution. Convergence of the primal variables does not necessarily imply the convergence of the duals. This leads to a clarification of the Covector Mapping Theorem in its relationship to the convergence properties of PS methods and its connections to constraint qualifications. Conditions for the convergence of the duals are described and illustrated. An application of the ideas to the optimal attitude control of NPSAT1, a highly nonlinear spacecraft, shows that the method performs well for real-world problems.