About one control problem described by system of inteqro-diferential equations

Consider the problem of minimizing the functional of terminal type S (v) = J ф(х, z(t₁, x))dx , (1) ^x0 with constraints ^zt = Д^xzyX (X^{x) e}D = ^t1^{] x} ^^xl^], (2) x, y(t, x) = J g (t, x, s, z(t, s))ds, x e[ x₀, xj, ^x0 _x (3) z(t₀, x) = J F (x, s, z(t₀, s), v(s))ds. ^x0 Here f (t, x, z, y), g(t, x, s, z) is the given n- and m-dimensional vector functions respectively, continuous with respect to all the variables together with partial derivatives of the vectors of state, F (x, z, v) is the given n-dimensional vector-function continuous with respect to all the variables together with partial derivatives z, t₀, t₁, x₀, x₁ (t₀ ^ /-J; xQ ^ x1 ^ are given, ф(x, z) is a scalar function continuous with respect to all the variables together with the ^x, z), v = v (x) is piecewise continuous (with a finite number of points dz of discontinuity of the first kind) vector control actions with values from a specified non-empty and bounded set that is, v(x) eV с R^r, x e [x₀,x₁]. Our goal is to derive a necessary optimality condition in the problem under consideration.

Keywords

Authors

References