Necessary conditions for first- and second-order optimality in ordinary dynamic systems with delay under multipoint functional constraints of the equalities and inequalities type

The article considers one optimal control problem described by a system of differential equations with a retarded argument in the presence of multipoint functional constraints such as equalities and inequalities. The intro-duction provides an overview of works related to the topic of the article. The first section is devoted to the formulation of the optimal control problem. In the second section, the necessary condition for optimality is proved in the form of an analogue of the Euler equation. In the third section, the general necessary condition for second-order optimality is formulated in terms of second variations of functionals defining constraints and quality criteria. From it, a necessary condition for second-order optimality was obtained, explicitly expressed through the parameters of the optimal control problem under consideration, and having a constructive nature. An analogue of the Legendre-Clebsch condition was established. Next, we study the case of a terminal quality criterion in the presence of terminal functional equations of the type of equalities and inequalities. All necessary conditions for first- and second-order optimality are established without assumptions of normality.