| Abstract: |
| In optimal control theory one sometimes extends the minimization domain of a given problem, with the aim of achieving the existence of an optimal control. However, this issue is naturally confronted with the possibility of a gap between the original infimum value and the extended one. Avoiding this phenomenon is not a trivial issue, especially when the trajectories are subject to endpoint constraints. Since the seminal works by J. Warga in 1970s, some authors have recognized `normality` of an extended minimizer as a condition guaranteeing the absence of an infimum gap. (Let us recall that an extremal is called abnormal provided the corresponding cost multiplier in the Maximum Principle can be chosen equal to zero, and normal otherwise.)
In particular, in 2020 Palladino and Rampazzo proposed a generalization of Warga`s criterion to a vast class of endpoint-constrained minimum problems` extensions through the combined use of the notion of abundance (due to Warga and Kaskosz) and of a suitable set separation theorem.
Yet, normality is far from being necessary for this goal, a fact that makes the search for weaker assumptions a reasonable aim. In relation with a control-affine system with unbounded controls, we provide a sufficient no-gap condition based on a notion of higher order normality, which is less demanding than the standard normality and involves iterated Lie brackets of the vector fields defining the dynamics. |
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