Optimal control of time‐fractional Navier–Stokes equations with Rosenblatt process

Abstract This manuscript focuses on solving the nonlinear time‐fractional Navier–Stokes equations with Rosenblatt process and bounded delay in Hilbert space. Firstly, this work aims to reformulate the nonlinear partial differential equation using stochastic calculus, Stokes operator and Helmholtz–Hodge projection operator. By applying Mittag‐Leffler functions, Schauder's fixed point theorem and stochastic analysis, the existence of mild solution is obtained. Under suitable assumptions, optimal control results are obtained for the proposed system through Balder's theorem. Finally, numerical examples are presented with and without delay, which ensures the obtained theoretical results.