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A formal theoretical explanation of the model-mismatch instability problem associated with certain adaptive control design schemes is proposed, and a solution is provided. To address the model-mismatch problem, a primary task of adaptive control is formulated as finding an asymptotically optimal, stabilizing controller, given the feasibility of adaptive control problem. A class of data-driven cost functions called cost-detectable is introduced that detect evidence of instability without reference to prior plant models or plant assumptions. The problem of designing adaptive systems that are robustly immune to mismatch instability problems is thus placed in a setting of a standard optimization problem. We call the result safe adaptive control because it robustly achieves adaptive stabilization goals whenever feasible, without prior assumptions on the plant model and, hence, without the risk of model-mismatch instability. The result improves the robustness of previous results in hysteresis switching control, both for discrete and for continuously-parameterized candidate controller sets. Examples are provided.