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An asymptotic tracking control law is proposed for a class of strict-feedback nonlinear systems with unknown nonlinearities. A Barrier Lyapunov function in combination with backstepping is proposed to guarantee that the output trajectory is contained in a predefined set. A single neural network (NN), whose weights are tuned online, is utilized in our design to approximate the unknown functions in the system dynamics, while the singularity problem of the control gain function is avoided. Meanwhile, in order to compensate for the NN residual reconstruction error and system uncertainties, a robust term is introduced and asymptotic tracking stability is achieved. All the signals in the closed-loop system are proved to be bounded via Lyapunov synthesis and the output converges to the desired trajectory asymptotically without transgressing a given bound. Finally, the merits of the proposed controller are verified in the simulation environment.