On the nontriviality of the maximum principle for control problems with state constraints

Versions of the Pontryagin maximum principle are available, which allow for a unilateral state constraint. They assert existence of multipliers, with respect to which an optimal control satisfies various conditions. Interest has recently focused on problems for which the state constraint is active at the initial time. For such problems these optimality conditions are degenerate: a trivial choice of multipliers is always possible, which conveys no useful information about optimal controls. The authors present new non-degenerate conditions for this situation. It is shown that, under a rather natural constraint qualification, choices of multipliers may be made, besides the trivial ones