Integral necessary condition of optimality of the second order for control problems described by system of integro-differential equations with delay

We consider the optimal control problem that is described by the system of integro-differential equations of the Volterra type with delay and multipoint performance criterion. The first and the second variations of the performance criterion are calculated under the hypothesis that the control domain is open. The necessary condition of the first order optimality in the form analogous to the Euler equations is deduced from the equality of the first variation of performance criterion and zero along the optimal process. Next, the implicit necessary condition of the second order optimality is obtained, which helps to establish rather general but constructively verified necessary condition for the second order optimality. The obtained results are applicable for constructing easy-verifying necessary conditions of optimality for the singular (in the usual sense) controls