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In the class of stochastic differential systems of equations of Itô type with indirect control by the first or second derivative, we consider the inverse dynamic problem of constructing a system controller such that a given manifold is an integral manifold of the system. In both cases, the quasi-inversion method is used to construct the whole set of equations of controllers providing the solution of this problem. [ABSTRACT FROM AUTHOR]

By the method of Lyapunov functions, we establish sufficient conditions of stability and asymptotic stability in probability for the integral manifold of the Itô differential equations in the presence of random perturbations from the class of processes with independent increments. Theorems on stochastic stability of the analytically given integral manifold of differential equations are proved in the first approximation and under the permanent action of small (in the mean) random perturbations. [ABSTRACT FROM AUTHOR]