2-regularity and 2-normality conditions for systems with impulsive controls

In this paper a controlled system with impulsive controls in the neighborhood of an abnormal point is investigated. The set of pairs (u,α) is considered as a class of admissible controls, where u is a measurable essentially bounded function and α is a finite-dimensional Borel measure, such that for any Borel set B, α(B) is a subset of the given convex closed pointed cone. In this article the concepts of 2-regularity and 2-normality for the abstract mapping φ, operating from the given Banach space into a finite-dimensional space, are introduced.