In this paper, we consider a distributed-order fractional stochastic differential equation driven by Lévy noise. We, first, prove the existence and uniqueness of the solution. A Euler–Maruyama (EM) scheme is constructed for the equation, and its strong convergence order is shown to be $min\{1-{\alpha }^{\ast },0.5\}$ , where ${\alpha }^{\ast }$ depends upon the weight function. Besides, we present a fast EM method and also the error analysis of the fast scheme. In addition, several numerical experiments are carried out to substantiate the mathematical analysis.

This study focuses on the development of a fuzzy-neural-network velocity sensorless control (FNNVSC) scheme including a nonlinear observer and a fuzzy-neural-network (FNN) controller for an n-link robot manipulator to achieve high-precision position tracking. This nonlinear observer is used to estimate joint velocities of the robot manipulator. Then, a four-layer FNN is utilized for the major control role without auxiliary compensated control, and the adaptive tuning laws of network parameters are derived in the sense of Lyapunov stability theorem to ensure the stable control performance. Numerical simulations of a two-link robot manipulator actuated by dc servomotors are given to verify the effectiveness and robustness of the proposed FNNVSC methodology. In addition, the superiority of the proposed control scheme is indicated in comparison with proportional-differential control (PDC), Takagi-Sugeno-Kang (TSK)-type fuzzy-neural-network control (T-FNNC) and robust-neural-fuzzy-network control (RNFNC) systems.