INTEGRAL AND MULTIPOINT NECESSARY OPTIMALITY CONDITIONS OF QUASI-SINGULAR CONTROLS IN ONE OPTIMAL CONTROL PROBLEM.

Consider the minimum functional problem S(u, v) = phi(y(x(1))) + integral(t1)(t0) G(x, z(t(1), x))dx, under restrictions u(t) is an element of U subset of R-r, t is an element of T = [t(0), t(1)], v(x) is an element of V subset of R-q, x is an element of X = [x(0), x(1)], partial derivative z/partial derivative t = f(t, x, z, u), (t, x) is an element of D = T x X, z(t(0), x) = y(x), x is an element of X, (y) over dot = g(x, y, v), x is an element of X, y(x(0)) = y(0). Here phi(y), (G(x, z)) are continuous and twice continuously differentiable with respect to y,. z. scalar functions, U and V are the given non-empty bounded and convex sets, f(t, x, z, u), (g(x, y, v)) are n-dimensional vector-functions continuous in the aggregate of variables together with partial derivatives with respect to (z, u) (y, v)) up to second order inclusive, u(t), (v(x)) are the piecewise continuous (with a finite number of break points of the first kind) vectors of control actions, y(0) is a constant vector. The necessary optimality condition in the form of the linearized maximum principle is proved, and the necessary optimality conditions for quasi-singular controls are established.