On Controllability of an Implicit Stochastic Differential Equation With Rosenblatt Process and Non Instantaneous Impulses

ABSTRACT In this paper, we examine a class of implicit fractional stochastic differential equations that incorporate non‐instantaneous impulses and Poisson jumps, influenced by the Rosenblatt process, which captures long‐range dependence and memory effects. The implicit formulation adds significant analytical complexity, requiring the use of sophisticated mathematical tools. To address this, we establish the existence and uniqueness of solutions by employing the fixed‐point theorem, in conjunction with the theory of sectorial operators and techniques from stochastic analysis. In addition, we investigate various notions of Ulam's stability, which measure the sensitivity of solutions to initial perturbations and ensure the reliability of the system under small changes. The concept of controllability is also explored, demonstrating that under appropriate conditions, the system can be guided to a desired final state. To illustrate the theoretical results and enhance understanding, two detailed examples are provided, confirming the effectiveness and applicability of the proposed model.