Nondegenerate Necessary Conditions for Linear Optimal Control Problems with Higher Index State Constraints

There are certain optimal control problems with state constraints for which the standard versions of the necessary conditions of optimality are unable to provide information to select minimizers. In the recent years, there has been some literature on stronger forms of the maximum principle, the socalled nondegenerate necessary conditions, that are informative for those problems. These conditions can be applied when certain constraint qualifications are satisfied. However, when the state constraints have higher index (i.e. their first derivative with respect to time does not depend on the control) the nondegenerate necessary conditions existent in the literature cannot be applied. This is because their constraint qualifications are never satisfied for higher index state constraints. Here, we provide a nondegenerate form of the necessary conditions that can be applied to problems with higher index state constraints. We note that control problems with higher index state constraints arise frequently in mechanical systems, when there is a constraint on the position (an obstacle in the path, for example) and the control acts as a force or acceleration.