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In this paper, we consider global adaptive control of nonlinearly parametrized systems in parametric-strict-feedback form. Unlike previous results, we do not require a priori bounds on the unknown parameters, which is as in the linear parametrization case. We also allow unknown parameters to be time-varying provided they are bounded. Our proposed adaptive controller is a switching type controller, in which the controller parameter is tuned in a switching manner via a switching logic. Global stability results of the closed-loop system have been proved.