Gap phenomena and controllability in free end-time problems with active state constraints

This paper is concerned with gap phenomena and controllability conditions for free end-time optimal control problems with endpoint and state constraints, in which the data are permitted to be measurable with respect to the time variable. In particular, we prove sufficient conditions to avoid a gap between the infimum of the original minimum problem and an extended problem, obtained by first enlarging the set of original controls and then convexifying the extended velocities set. These conditions, which also guarantee controllability of the original system to an extended solution, are given in terms of normality of multipliers for the Maximum Principle, involving an extended minimizer with possibly active state constraint at the endpoints. In the free time case, links between absence of a gap and normality have only recently been studied, for the relaxed problem without state constraints. This paper establishes such links for a more general extension admitting active state constraints. Furthermore, under additional constraint qualification conditions we improve the normality test for no gap, by considering nondegenerate multipliers only.