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An adaptive compensation control scheme using output feedback is designed and analysed for a class of non-linear systems with state-dependent non-linearities in the presence of unknown actuator failures. For a linearly parameterized model of actuator failures with unknown failure values, time instants and pattern, a robust backstepping-based adaptive non-linear controller is employed to handle the system failure, parameter and dynamics uncertainties. Robust adaptive parameter update laws are derived to ensure closed-loop signal boundedness and small tracking errors, in general, and asymptotic regulation, in particular. An application to controlling the angle of attack of a non-linear hypersonic aircraft dynamic model in the presence of elevator segment failures is studied and simulation results show that the developed adaptive control scheme has desired actuator failure compensation performance.