An Analogue of the Euler Equation and Necessary Conditions for Second-order Optimality in an Optimal Control Problem Described by Nonlinear Volterra Integral Equation

The optimal control problem of the minimum of a multipoint functional defined on solutions of a nonlinear integral equation is considered, and implicit necessary conditions for optimality of the first and second orders are obtained. Also using them, an analogue of the Euler equation was established and constructively verifiable necessary conditions for second-order optimality were obtained. Singular optimality controls in the classical sense have been studied.