On necessary optimality conditions for discrete control systems

In this article, under weakened assumptions, we study high-order necessary optimality conditions for discrete optimal control problems with the free right end of the trajectory. Here, we first use the concept of the relative interior of a set in the broad sense, and then the combination of linear (i.e. uniformly small) and needle variation of the admissible control. As a result, a new formula for the increment of the quality functional with the members of zeroth, first and second order of smallness is obtained. This formula serves as a source of the well-known zeroth order necessary optimality condition, if the admissible control has no linear variation, or the well-known first and second order necessary optimality conditions, if the increment of the quality functional of order zero is vanished on a certain subset of the domain of admissible controls. Following the obtained formula of the increment of the quality functional, the concepts of zeroth, first and second variations of the quality functional are introduced in a more general form, from which, in particular, the well-known variations of the quality functional follow. Based on the obtained formulae for the variations of the quality functional, using the needle variation of the admissible control, more constructive the zeroth, first and second order necessary optimality conditions with broad applications area are obtained.