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In this paper, adaptive neural control is investigated for a class of unknown multiple-input multiple-output nonlinear systems with time-varying asymmetric output constraints. To ensure constraint satisfaction, we employ a system transformation technique to transform the original constrained (in the sense of the output restrictions) system into an equivalent unconstrained one, whose stability is sufficient to solve the output constraint problem. It is shown that output tracking is achieved without violation of the output constraint. More specifically, we can shape the system performance arbitrarily on transient and steady-state stages with the output evolving in predefined time-varying boundaries all the time. A single neural network, whose weights are tuned online, is used in our design to approximate the unknown functions in the system dynamics, while the singularity problem of the control coefficient matrix is avoided without assumption on the prior knowledge of control input's bound. All the signals in the closed-loop system are proved to be semiglobally uniformly ultimately bounded via Lyapunov synthesis. Finally, the merits of the proposed controller are verified in the simulation environment.