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An adaptive output feedback neural network controller is designed, which is capable of rendering affine-in-the-control uncertain multi-input-multi-output nonlinear systems strictly passive with respect to an appropriately defined set. Consequently, a simple output feedback is employed to stabilize the system. The controlled system need not be in normal form or have a well-defined relative degree. Without requiring a zero-state detectability assumption, uniform ultimate boundedness, with respect to an arbitrarily small set, of both the system's state and the output is guaranteed, along with boundedness of all other signals in the closed loop. To effectively avoid possible division by zero, the proposed adaptive controller is of switching type. However, its continuity is guaranteed, thus alleviating drawbacks connected to existence of solutions and chattering phenomena. Simulations illustrate the approach.