An analog of the Euler equation and necessary optimality conditions of the second order in one optimal control problem with variable structure

The article considers an optimal control problem with variable structure, described by a combination of differential and integral equations, as well as by a performance functional of terminal type. Control fields are open. We have proved implicit necessary optimality conditions of the first and second orders. An analog of the Euler equation and an analog of Legendre-Clebsch condition are proved on the basis of studying necessary optimality conditions. The obtained new sequences of multipoint necessary conditions for the optimality of classical singular controls allow us to narrow down the set of admissible controls suspicious for optimality.