When Is the Maximum Principle for State Constrained Problems Nondegenerate?

It is well known that a form of the Pontryagin maximum principle applies to optimal control problems involving a unilateral state constraint. We discuss how the need arises to solve problems of this nature when the left endpoint is fixed and when the initial state lies in the boundary of the state constraint set region. In such cases, previous versions of the maximum principle convey no information, since a trivial choice of multipliers may be made, namely one in which the state constraint multiplier is a unit measure concentrated at the left endtime and all the other multipliers are zero. We prove strengthened forms of the maximum principle applicable to such situations: constraint qualifications are formulated under which multipliers exist, besides the trivial ones.