Optimal control for fractional stochastic integrodifferential system driven by Rosenblatt process with Poisson jumps

This article aims to investigate the optimal control of a class of fractional stochastic integrodifferential equations (SIDEs) driven by a Rosenblatt process and Poisson jumps in a Hilbert space. Fractional-order derivatives offer the advantage of more accurately modelling real-world scenarios compared to integer-order derivatives, particularly in problems involving memory or hereditary characteristics. Despite the widespread belief that the Rosenblatt process is insufficient and cannot be substituted by fractional Brownian motion (fBm), we show that the Rosenblatt process does, in fact, converge and can be used to real-world scenarios. We provide a detailed discussion of the Rosenblatt process, including its properties, and an explanation of how it can be approximated by the Wiener process. We study a novel set of sufficient conditions for the existence and uniqueness of mild solutions. Then the optimal control is demonstrated through the application of Balder's theorem. Finally, we validate the theoretical findings with a practical example and a numerical simulation.