It has been observed elsewhere in the literature that the activation of constraints in a trajectory optimization problem can lead to higher index DAEs. Several existing codes can handle a number of these constrained problems. In this note we will discuss why the situation is more complex than just saying a higher index DAE occurs. The discussion is in the context of a specific industrial code SOCS but the observations made here have relevance for a number of methods and have implications for what types of test problems a code should be tested on.
Biehn, N., Campbell, S. L., Jay, L., & Westbrook, T. (2000). Some comments on DAE theory for IRK methods and trajectory optimization. Journal of Computational and Applied Mathematics, 120(1), 109–131. https://doi.org/10.1016/S0377-0427(00)00306-X