Gap Phenomena in Optimal Control with State Constraints

When existence of minimizers of an optimal control problem is not guaranteed, it is a common practice in Control Theory to extend the set of admissible solutions, so that to construct an auxiliary optimization problem that admits minimizers. The first fundamental requirement of such an auxiliary problem for it to be well posed is the density (e.g. in the L∞-norm) of the set of trajectories of the original system into that of the auxiliary one. Nevertheless, due to the presence of constraints, it might happen that the minimum of the auxiliary problem is strictly smaller than the infimum of the original one. We refer to this phenomenon as infimum gap. In the literature, sufficient conditions for no gap are sometimes expressed in terms of normality of the sets of multipliers of the Maximum Principle. However, in the common situation of active state constraints at the initial point, there always exist degenerate – consequently, abnormal – sets of multipliers. In this thesis, we establish a gap-abnormality relation for a general auxiliary prob- lem that comprehends as special cases both the compactification of the control set and the convexification of the dynamics in a novel unified framework. Furthermore, we provide refined no infimum gap conditions in order to deal with the presence of state constraints. In particular, under a suitable constraint qualification condition, we prove that if the minimizer of the auxiliary problem is a nondegenerate normal extremal, i.e. it is normal in the subset of nondegenerate multipliers only, then there is no infimum gap. We highlight the relevance and novelties of our results with several examples, and we analyze in detail the special case of control-polynomial impulsive optimization problems.