Abstract: To address the problem of quantitatively constraining overshoot in quadrotor unmanned aerial vehicles (UAVs) under time-varying disturbances, we propose a neural network adaptive control method with prescribed performance based on a novel time-varying barrier Lyapunov function(BLF). First, the overshoot constraint problem is analyzed, and a new asymmetric time-varying BLF is designed to impose continuous constraints and enhance the flexibility of the performance boundary. Second, a tubular prescribed performance function is constructed to enforce quantitative overshoot limits and meet steady-state performance requirements. Using the backstepping method, a feedback control law and a neural network adaptive law are developed to ensure that system performance constraints are satisfied. Stability analysis proves that all closed-loop signals are uniformly ultimately bounded. Simulation results confirm that the proposed controller effectively constrains overshoot and ensures robust, high-accuracy tracking. The proposed method is particularly applicable in realistic scenarios, such as navigating narrow passages or carrying suspended load, where overshoot constraints are critical. In the realm of contemporary control methodologies, while extensive research has been dedicated to regulating system overshoot, the prevailing approach for adjusting transient performance predominantly relies on parameter tuning. Such parameter-based strategies, albeit widely adopted, often lack a systematic mechanism to enforce rigorous bounds on overshoot magnitudes. Notably, a significant gap persists in the literature regarding the realization of quantitative constraints on overshoot, which is critical for ensuring predictable system behavior in high-precision engineering applications. In recent years, several scholars have proposed a dynamic tube-based Model Predictive Control (MPC) framework. Within this framework, system states are confined to a predefined tube; meanwhile, the geometric structure of the tube is designed, and a sliding mode controller is employed to impose constraints on system variables. Nevertheless, the framework fails to address the constraint of overshoot, and the dynamic tube it constructs lacks inherent binding force. Therefore, in this paper, drawing on the barrier Lyapunov function theory, this study establishes a set of tubes with binding force. By predefining overshoot constraints via geometric configurations, quantitative constraints on the system overshoot are ultimately achieved. The radial basis function neural network is employed to estimate multi-source time-varying disturbances, and its adaptive law ensures effective disturbance rejection. Comparison experiments show that the control strategy restricts system errors within a predefined tubular region and outperforms conventional methods in overshoot reduction. Furthermore, the method allows for the design of both transient and steady-state performance in advance, thereby eliminating the need for repeated parameter tuning.