Quasisingular control in a one step control problem of discrete two-parametric systems

We study one stepwise (i.e., multi-stage) optimal control problem of a terminal type by a quality functional, described by discrete two-parameter systems of equations of the Fornasini-Marchezini type under the assumption of convexity of the control domains. A discrete two-parameter system of equations of the Fornasini-Marchezine type is a difference analogue of the system second-order hyperbolic equations (sometimes such systems of equations in the Western literature are also called 2D systems). Using a modified analogue of the increment method, a special decomposition of the second-order quality functional, using linearized difference systems of equations is obtained. Using one version of the increment method, the first-order necessary optimality condition is established in the form of a linearized (differential) maximum condition. The case of degeneration of the linearized maximum condition (a quasi-singular case) separately is studied. Using constructive verifiable quadratic necessary optimality conditions for quasi-singular controls, using representations of solutions of linearized difference systems of equations using special formulas for incrementing the quality functional.