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An adaptive failure compensation control scheme is developed based on linear quadratic (LQ) control for discrete-time possibly nonminimum phase systems with unknown actuator failures with characterizations that some unknown system inputs are stuck at some unknown fixed or varying values at unknown time instants. The failure compensation control problem is formulated as a new LQ control problem. Based on some results on new controller structures and design conditions developed in [9], a stable adaptive law is proposed for the estimation of uncertain parameters from both the plant and actuator-failures. The adaptive failure compensation controller is implemented with adaptive parameter estimates, for achieving closed-loop stability and output regulation. Simulation results are presented to verify the desired adaptive control performance in the presence of uncertain actuator failures.