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A constrained approximate dynamic programming (ADP) approach is presented for designing adaptive neural network (NN) controllers with closed-loop stability and performance guarantees. Prior knowledge of the linearized equations of motion is used to guarantee that the closed-loop system meets performance and stability objectives when the plant operates in a linear parameter-varying (LPV) regime. In the presence of unmodeled dynamics or failures, the NN controller adapts to optimize its performance online, whereas constrained ADP guarantees that the LPV baseline performance is preserved at all times. The effectiveness of an adaptive NN flight controller is demonstrated for simulated control failures, parameter variations, and near-stall dynamics.