About one discrete two-parameter control problem

We consider a discrete optimal control problem described by a system of two-dimensional Volterra-type difference equations with a controlled initial condition. It is assumed that the initial function is a solution to the one-dimensional Volterra difference equation. The quality functionality is terminal. The process is controlled with two classes of control functions. Control areas are bounded and open sets. The first and second (in the classical sense) variations of the quality functional are calculated. An analogue of the Euler equation is proved and a number of con-structively verifiable necessary conditions for second-order optimality are established.