A numerical approach of trajectory controllability for nonlinear stochastic systems with an application

This article is devoted to examine the trajectory controllability of nonlinear stochastic differential equations. The existence and uniqueness of  solution are formulated and proved by utilizing generalized Banach fixed point theorem. Moreover, providing adequate assumptions, trajectory controllability for nonlinear stochastic system is established by imposing Gronwall’s inequality. At last, some numerical examples are presented to verify the established theoretical concepts. To address the problem on a surface, an Euler-Lagrange optimization problem is considered, which is aiming to determine the shortest trajectory on the surface. A numerical simulation is also illustrated to showcase the effectiveness of the algorithm developed for the proposed model which is unique and makes this work more interesting.