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In this paper, a class of time-space fractional stochastic delay control problems with fractional noises and Poisson jumps in a bounded domain is considered. The proper function spaces and assumptions are proposed to discuss the existence of mild solutions. In particular, approximate strategy is used to obtain the existence of mild solutions for the problem with linear fractional noises; fixed point theorem is used to achieve the existence of mild solutions for the problem with nonlinear fractional noises. Finally, the approximate controllability of the problems with linear and nonlinear fractional noises is proved by the property of mild solutions.

Approximate Controllability of Fully Nonlocal Stochastic Delay Control Problems Driven by Hybrid Noises