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.