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Sets of accessibility for controlled stochastic differential equations

1. For ordinary differential equations a measurable function of time is usually called an admissible control, since an attempt to choose a control depending on the phase trajectory (so-called synthesis) comes up against the problem of existence and uniqueness of a solution. In the stochastic case, when the diffusion matrix is non-singular, there is an alternative: either consider the class of admissible controls compatible with a preassigned flow of σ-algebras (in this case there is a strong solution under the traditional conditions on the coefficients), or, as the admissible controls, consider functionals compatible with the flow generated by the control process itself, that is, depending on the phase trajectory. In the latter case the existence of a strong solution cannot be guaranteed, but for a wide class of coefficients there is a weak solution, which is, moreover, unique [1], [3]. In the present note we shall state a result that establishes a connection between the sets of accessibility in the space of distributions corresponding to the two approaches above. For the case of unit diffusion it turns out that if the convexity condition (R) is satisfied the set of accessibility in the scheme of strong solutions is densely embedded in the set of accessibility in the scheme of weak solutions, but in general these sets do not coincide. The proof, and the application to control problems for singularly perturbed linear stochastic differential equations, are given in [2],