This paper is devoted to analysing a kind of fractional neutral stochastic system (FNSS). Firstly, by introducing the notion of newly defined two-parameter Mittag-Leffler matrix function, we derive the solution of the corresponding linear stochastic system. Subsequently, for the linear case, by virtue of the Grammian matrix, we give a suffcient and necessary condition to guarantee the relatively exact controllability for the addressed case. Furthermore, for the nonlinear one, the relatively exact controllability is obtained by fixed point and explore it via Banach contraction principle. Finally, two examples are provided to intensify our theoretical conclusions.

In this article, we consider the relatively exact controllability of fractional stochastic delay system (FSDS) driven by Lévy noise. Firstly, we derive the solution of linear FSDS via delayed matrix functions of Mittag-Leffler (M-L). Subsequently, by virtue of the controllability Grammian matrix, we explore the relatively exact controllability of linear FSDS. In addition, with the aid of Jensen’s inequality, Hölder’s inequality and Itô’s isometry, the existence and uniqueness of the considered nonlinear FSDS are investigated by employing Banach contraction principle.Thereafter, the relatively exact controllability of nonlinear FSDS is discussed. Finally, the theoretical results are supported through an example.

In this paper, a class of stochastic fractional differential equations (SFDEs) with impulses is considered. By virtue of the Monch’s fixed point theorem and Banach contraction principle, we explore the existence and uniqueness of solutions to the addressed system. Furthermore, with the aid of the Jensen’s inequality, Holder inequality, Burkholder-Davis-Gundy inequality, Gronwall-Bellman inequality and some novel assumptions, the averaging principle of our considered system is obtained. At the end of this paper, an example is provided to illustrate the theoretical results.