Abstract
In this paper we consider the problem of the existence of optimal control systems described by the stochastic Ito differential equation. It is shown (Theorem 1) that Fleming’s existence theorem [1, Thm. 3, p. 205] remains valid without the assumption that the drift coefficient f of the system is linear in the control variable. Further, it is shown that the control restraint set U can be taken as variable. Our result is based on the Fillippov technique (Himmelberg et al. [2, Thm. 3, p. 281]) rather than the lower semicontinuity arguments as used by Fleming [1, 1968, Appendix 3, p. 213]. However, our result does not contain his.

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