Existence and stability behavior of fractional stochastic differential equation driven by Rosenblatt process

This work focus on a nonlinear mathematical model deal with the stability of visual trajectory track during locomotion in the fish robot is developed. Previously the visual trajectory model has derived through stochastic differential equation. The source of long memory, and more precisely of infinite memory, is due to infinitely large time constants. Thus, Fractional calculus theory is developed to get better model. (1) The fractional stochastic differential equation (FSDEs) is utilized to determine the parameters that ensure the coordination between the unstable visual trajectory track and the stable visual trajectory track of the fish robotic system and robot driver. (2) Existence and stability results are derived through successive approximation and Bihari’s inequality, semigroup theory, and fractional calculus in stochastic settings. (3) Stability results of Rosenblatt process and numerical simulation are established and applied for collision free track in the visual trajectory track of the fish robot. (4) Stability of FSDEs through Rosenblatt process entrusted depletion of collision in the ocean water environment even in tiny particles from the visual trajectory to the fish robot. There is no existing knowledge in this regard. Therefore, the study is conducted. (5) The algorithms have several advantages from gaze shift frame such as terrific quality of randomness, key sensitivity, and collision free location stability. Numerical simulation results manifest the effectiveness, efficiency and feasibility of real-world applications.