On optimality of singular controls in control problem of the step discrete two-parametric systems

X), x = x₀, x₀ +1,..., X, z(t,x₀) = P1 (t), t = to,to + 1,...,t1, (4) ^a (^xo ) = P1 (^to )^, y (t +1,x +1) = g(t,x,y (t,x),v(t,x)), (t,x) e D , (5) y (t_x, x) = G (x, z (tj, x)), x = x₀, x₀ +1,..., X, y(t,xo) = P2 (t), t = t1,t1 +1,...,t₂, (6) ^G (^xo^, z (^tl^{, x}o ))= P2 (^t1 ). Here f (t, x,z,u), (g (t,x,y,v)) is a given n (m)-dimensional vector function that is continuous with respect to the set of variables together with its partial derivatives with respect to z (y) up to the second order inclusive, ф1(г), ф2(у) are given twice continuously differentiable scalar functions, a(x), в (t), i = 1,2 are given discrete vector-valued functions of corresponding dimensions, u (t,x) (v(t,x)) is r (q)-dimensional control actions vector, U, Vare given non-empty and bounded sets, G(x,z) is a given m-dimensional vector-valued function continuous with respect to the set of variables together with its partial derivatives with respect z up to second order inclusive, t₀, ^, t₂, x₀, X are given numbers, and the differences t, -1₀ and X - x₀ are integers. The first order necessary optimality conditions of the Pontryagin maximum principle type is established and singular case is investigated.

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