Optimal control of time-fractional stochastic Burgers' equation driven by mixed fractional Brownian motion

This paper demonstrates the existence, uniqueness, and optimal control of the time -fractional stochastic Burgers' equation with a deviated argument governed by mixed fractional Brownian motion. The nonlinear deterministic controlled time -fractional Burgers' equation is subjected to stochastic perturbations to model the memory effects and disturbances that arise in the turbulent flow of liquids. The solvability of the presented system is investigated using stochastic analysis, fractional calculus, and the Nussbaum fixed point theorem. Balder's theorem is also used to prove the existence of optimal control. At last, an example is presented to demonstrate the developed theory.