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In this paper, we show, for any nonlinear system that is asymptotically controllable to a compact set, that a logic-based, hybrid feedback can achieve asymptotic stabilization that is robust to small measurement noise, actuator error, and external disturbance. The construction of such a feedback hinges upon recasting a stabilizing patchy feedback in a hybrid framework by making it dynamic with a discrete state, while insisting on semicontinuity and closedness properties of the hybrid feedback and of the resulting closed-loop hybrid system. The robustness of stability is then shown as a generic property of hybrid systems having the said regularity properties. Auxiliary results give uniformity of convergence and of overshoots for hybrid systems, and give a characterization of asymptotic stability of compact sets.