About one A.I. Moskalenko control problem

Suppose you want to minimize the multi-point functional xj S (u ) = cp(y (X1), y (X 2),..., y (X_k )) + J F (x, z (1, x), z (72, x),..., z (, x))dx (1) ^x with constraints u(t)е U с R^r, t е T = [t₀,t₁ ], v(x)gV сR^q , x е X = [x₀,xj ], (2) ^^ = f(t, x, z (t, x), u(t)), (t, x )е D = T x X , (3) z(t₀, x)= y(x), x e X , (4) ^d- = g(x,y(x),v(x)), x e X , (5) dx y^(xo ⁾ = Уо · ⁽⁶⁾ Here f (t, x,z,u) (g(x,y,v)) is a given и -dimensional vector-function defined and continuous in D x Rⁿ x R^r (x x Rⁿ x R^q) together with partial derivatives with respect to z (y) up to the second order inclusive, T_i e(t₀, t₁ ] i = 1, k, ((₀ < T₁ < T₂ < ... < T_k < t₁), X_i e (x₀,x₁ ] (x₀ < X₁ < X₂ < ... < X_к < x₁) are given points, ф(a₁,a₂a_k) (F (x, b₁, b₂b_k )) is a given and continuous in R^k(X x R^k) together with the partial derivatives up to the second order inclusive a scalar function, U and V are non-empty and bounded sets, u (t) (v (x)) is a piecewise continuous (with a finite number of points of discontinuity of the first kind) vector-function of control actions. Paper deals obtaining the necessary conditions for optimality in the problem (1)-(6).

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