First and second necessary optimality conditions for a discrete optimal control problem with nontransitive performance index

The paper deals with a nonlinear discrete optimal control problem with a vector-valued performance index of terminal type. The main difference between the problem studied here and similar problems considered in other works is that the preference relation used here for comparison of admissible controls is not assumed to be in general transitive. We assume that a preference relation is only an asymmetric binary relation compatible with algebraic operations defined on a space of vector estimations. It is shown that under such assumptions the optimal control problem in question can be reduced to the problem of minimizing a scalar function, that is constructed as a composition of a difference sublinear function representing the preference relation and the vector performance index of the initial problem, over the set of admissible trajectories of the discrete system. The vector performance index is assumed to be twice parabolic differentiable and, hence, it is nonsmooth whereas the discrete system satisfy traditionally assumptions of smoothness. Under such assumptions we analyze the reduced scalar optimal control problem with variational tools and derive in such a way first and second necessary optimality conditions for admissible controls of the initial problem which extend such classical optimality conditions as the Euler condition and the nonnegativity of second variation are proved.