On optimal quasi-singular controls in stochastic control problem

In the report we consider a stochastic optimal control problem whose mathematical model is given by stochastic differential equation Ito. Let (Omega, F, P) be a complete probability space. w(t) be n dimensional standard Winer process determined on the space (Omega, F, P). L-F(2) (t(0), t(1): R-n) be a space of measurable with respect by (t,omega) random processes x(t,omega): [t(0), t(1)]: Omega -> R-n such that E (t1)integral(t0) parallel to x (t)parallel to(2) dt < +infinity. Here and follows sign E is mathematical expectation. Let on a fixed time interval. [t(0), t(1)]= T the control process be described by the following stochastic differential system: dx (t) = f (t, x(t), u(t))dt+sigma (t,x(t))dw(t), t is an element of T. x(t(0)) = x(0). Here f(t, x, u) is the given n dimensional vector-function continuous in totality of variables together with partial derivatives with respect by (x, u) to second order inclusively, sigma(t, x) is a matrix function of sizes (n x n) continues in totality of variables together with partial derivatives with respect by x to second order inclusively. u (t) is an element of U-d equivalent to {u(.) is an element of L-F(2) (t(0), t(1); R-r)vertical bar u(t) is an element of U subset of R-r}, where U is the given nonempty, bounded and convex set. Call U-d set of admissible controls. Our goal by minimize the functional I(u) = E{h(x(t(1)))}, on the set of admissible controls. Here h(x) is the given twice continuously differentiable scalar function. By means of the stochastic analogue of the method suggested and developed in the papers of K.B. Mansimov, we get a linearized necessary optimality condition, and also study the quasi-singular case. Necessary optimality condition of quasi-singular controls is established. Then investigated particular cases.