This paper presents an extension of the L1 adaptive controller to a class of nonlinear systems where the control effectiveness is time-varying and unknown, but with a known sign. Moreover, this class of nonlinear systems contains time-varying and unknown state-dependent nonlinearities. The proposed L1 adaptive controller consists of three components, a state predictor used to estimate real states, an adaptive law used to update the adaptive parameters in the state predictor, and a low-pass filtered control law. First, the stable closed-loop reference system is constructed. Then, the estimation errors between estimated states and real states are proved to be arbitrarily small by increasing the adaptation rate. After that, we further prove that the adaptive controller ensures uniformly bounded transient and asymptotical tracking of the reference system. The performance bounds can be systematically improved by increasing the adaptation rate. Simulation results on a single-link nonlinear robot arm verify the theoretical findings.