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, μ) is considered as a class of admissible controls, where u is a measurable essentially bounded function and μ is a finite-dimensional Borel measure, such that for any Borel set B, μ(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. The concepts of 2-regularity and 2-normality play a great role in the course of derivation of the first and the second order necessary conditions for the optimal control problem, consisting of the minimization of a certain functional on the set of the admissible processes. These concepts are also important for obtaining the sufficient conditions for the local controllability of the nonlinear systems. The convenient criterion for 2-regularity along the prescribed direction and necessary conditions for 2-normality of systems, linear in control, are introduced in this article as well.