An adaptive neural network prescribed performance control strategy is proposed based on Hamilton-Jacobi Inequality (HJI) theory with the background of manipulator control. Firstly，the nonlinear transformation with a prescribed performance function is utilized to convert the tracking error to an unconstrained form，thereby allowing the trajectory tracking error to converge into a prespecified range at a user-specified convergence rate. Secondly，the backstepping technique is used to design the virtual control law based on the unconstrained tracking error. Further，according to the universal approximation characteristics of neural network，RBF neural network is used to approximate the model uncertainty. Finally，a novel prescribed performance control method is designed based on the HJI theory and the approximation provided by the RBF neural network. The Lyapunov function proves the stability of the trajectory-tracking closed-loop system in this paper，and the effectiveness of the proposed control method is verified in the simulation of a two-joint manipulator.