An Angular Open Mapping Theorem

The purpose of this paper is to provide a modified open mapping theorem, which also applies to maps which are not open, but whose intersection with a convex cone is relatively open. We give a constructive proof, which for the special maps considered here may be seen as an alternative to the indirect proof of Frankowska’s more general open mapping principle. Open mapping theorems are of utmost importance in both the theories of optimality and local controllability, as only they allow one to use local approximating cones to conclude that a trajectory lies on the boundary (or in the interior) of the attainable set. In the theory of optimal control, e. g. the Maximum Principle (e.g. [4, 5, 6]), one typically uses a very restrictive definition of tangent vectors to a set which leads to small approximating cones. This is closely related to the fact that in general the attainable set may have inward corners (even inward cusps). This makes it impossible to work in that case with the most general – and most intuitive – notion of tangent vectors to a set and simultaneously have them form a convex cone, which is also an approximating cone to the attainable set. However, to obtain strong sufficient conditions for (small-time) local controllability (STLC) about an equilibrium point one wants to employ the most general notion of tangent vectors to a set. (For example, one does not want to be restricted to tangent vectors generated by families of Pontryagin control variations, or those employed in the High Order Maximal Principle [4], but instead be able to also consider tangent vectors generated by, e.g., discretely parametrized families of control variations with an increasing number of switchings like those employed in [3].