Stability and controllability of ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi $$\end{document}-Caputo fractional stochastic differential systems driven by Rosenblatt process with impulses

Many real-life problem is mathematically modeled by differential equations, there are always intrinsic phenomena that are not taken into account and can affect the behavior of such a model. For example, external forces can abruptly change the model. Sometimes these changes begin impulsively at some points and remain active over certain time intervals. The models of this situation are created using non-instantaneous impulses. So, in this work, we consider a new class of non-instantaneous impulsive ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi $$\end{document}-Caputo fractional stochastic differential systems driven by the Rosenblatt process. Using fractional calculus, the Banach fixed point method, and semigroup theory, we investigated the existence of piecewise continuous mild solutions for the proposed system. Then the novel stability criteria for the considered system are obtained by using the impulsive Gronwall’s inequality. Moreover, we discussed the controllability results for the proposed system. Finally, a numerical example is provided to show the validity of the established approaches.