Necessary optimality conditions of the first and second order in a single-step control problem described by a difference equation and a Volterra-type integro-differential equation

A stepwise optimal control problem described by a set of difference and Volterra-type integro-differential equations and a Bolza functional is considered. Previously, similar problems were studied for the case of differential and ordinary difference equations. Assuming that the control domains are open, using a modified version of the increment method, the first and second variations of the quality functional are calculated. Using these variations, an analogue of the Euler equation and a number of constructively verifiable necessary optimality conditions of the second order are proved.