This work studies the trajectory tracking control problem for under-actuated autonomous underwater vehicles (AUVs). In order to solve the under-actuated problem, an error definition method is used, which transformed the trajectory tracking control problem into the prescribed performance tracking control problem. The continuous dynamics of the trajectory tracking error, for which the actuated velocities are virtual control inputs, are formulated. By using the backstepping, a virtual control velocity is designed to guarantee the prescribed performance. Then the dynamic of velocity tracking error is formulated and a cost function is given. The optimal control law is designed by ADP. By combining the backstepping and ADP, not only an optimal control law is desgined, but also the prescribed performance is guaranteed. Moreover, since there is no need to select performance functions at dynamic level, it makes the design of the control scheme more easily. The Lyapunov theory is used to demonstrate that all error variables and the under-actuated states are uniformly ultimately bounded (UUB). The effectiveness of the proposed control scheme is validated using simulation results.