Direct transcription solution of higher-index optimal control problems and the virtual index

Direct transcription methods are a popular approach for solving optimal control problems. They are sometimes able to solve problems that the classical numerical theory would suggest that they cannot solve. The dynamics of an equality constrained optimal control problem form a differential algebraic equation (DAE), and most numerical methods for solving these optimal control problems require solving this DAE for the states and algebraic variables, which would imply that the problem is unsolvable if this DAE is high index. We will demonstrate that direct transcription methods, which use a different approach, can successfully solve some higher index equality constrained optimal control problems and that the role of the index needs reinterpretation when using direct transcription methods. These results and observations have implications for software design and guidance provided to users.