Abnormality and Strict-Sense Minimizers That Are Not Extended Minimizers

We consider a constrained optimal control problem and an extension of it, in which the set of strict sense trajectories is enlarged. Extension is a common procedure in optimal control, used to derive necessary and sufficient optimality conditions for the original problem from the extended one, which usually admits a minimizer and has a more regular structure. However, this procedure fails if the two problems have different infima. It is therefore relevant to identify such situations. Following on from earlier work by Warga but adopting perturbation techniques developed in nonsmooth analysis, we investigate the relation between the occurrence of an infimum gap and abnormality of necessary conditions. For a notion of local minimizer based on control distance and an extension including the impulsive one, we prove that (i) a local extended minimizer which is not a local minimizer of the original problem and (ii) a local strict sense minimizer which is not a local minimizer of the extended problem, both satisfy the extended maximum principle in abnormal form. The main novelty is result (ii), as until now it had only been shown that a strict sense minimizer which is not an extended minimizer is abnormal for an `averaged version' of the maximum principle.