Controllability Analysis of Neutral Stochastic Differential Equation Using \(\psi \)-Hilfer Fractional Derivative with Rosenblatt Process

In this study, we examine the controllability of a neutral stochastic fractional equation of motion that incorporates the \(\psi \) $\psi$ -Hilfer fractional derivative and the Rosenblatt process. Utilizing the measure of non-compactness and the Banach contraction mapping, we derive insights into the existence and unique characteristics of the mild solution for the system. We establish the prerequisites for controllability for both linear and nonlinear systems and confirm the controllability of nonlinear cases using the Banach contraction principle. To illustrate our theoretical findings, we provide numerical examples that highlight the practical implications of our analysis. This combined theoretical and numerical approach enhances the understanding of controllability in neutral stochastic fractional equations with the \(\psi \) $\psi$ -Hilfer fractional derivative and Rosenblatt process.