A No Infimum-Gap Criterion

In optimal control theory one happens to extend the class of admissible processes, for instance when trying to establish the existence of a minimum. In optimal control theory it may happen to extend the class of admissible processes, for instance when one tries to establish the existence of a minimum. Though such extensions are preferably as small as possible -for instance, one might consider the closure of the set of controls in some suitable topology- it is well known that a gap between the infimum value of the original problem and the infimum value of the extended problem may occur, notably because of end-point constraints. Coupling a notion of abundant introduced by J. Warga to a set-separation argument (based on the notion of Quasi Differential Quotient), we establish a general `normality' criterion for avoiding infimum-gaps. On the one hand, we show that this criterion applies to two classical domains' enlargements: the `relaxation' of non-convex bounded control problems and `the impulsive closure' of unbounded control problems. On the other hand, it can be utilized in different kinds of problem extensions, as it is suggested in the last section.to investigate a connected question concerning the Maximum Principle.