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 (Q, F, P) be a complete probability space. w(t) be n dimensional standard Winer process determined on the space (Q, F, P). L<sup>2</sup>f (?0,R") be a space of measurable with respect by (t, ю) random processes х(t, ю): [/0, ]:Q^ R" such that t<sub>1</sub> 2 E J |\х (t)|| dt. <sup>t</sup>0 Here and follows sign E is mathematical expectation. Let on a fixed time interval [t<sub>0</sub>, t<sub>1</sub> ] = T the control process be described by the following stochastic differential system: dх (t) = f (t, х (t), u (t)) dt + о (t, х (t)) dw (t), t e T, X <sup>(t</sup>0 <sup>)</sup> = <sup>х</sup>0. Here ft^u) is the given n dimensional vector-function continuous in totality of variables together with partial derivatives with respect by (х, u) to second order inclusively, о (t, х) is a matrix function of sizes (" x ") continues in totality of variables together with partial derivatives with respect by х to second order inclusively. u (t) e U<sub>d</sub> = {u (•) e L<sup>2</sup><sub>f</sub> (t<sub>0</sub>, t<sub>1</sub>;R<sup>r</sup> )|u (t)e U с R<sup>r</sup>}, where U is the given nonempty, bounded and convex set. Call Ud set of admissible controls. Our goal by minimize the functional I (u ) = E {h (х (t1))}, on the set of admissible controls. Here h(х ) 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.