An optimal tracking control problem for a class of nonlinear systems with guaranteed performance and asymmetric input constraints is discussed in this paper. The control policy is implemented by adaptive dynamic programming (ADP) algorithm under two event-based triggering mechanisms. It is often challenging to design an optimal control law due to the system deviation caused by asymmetric input constraints. First, a prescribed performance control technique is employed to guarantee the tracking errors within predetermined boundaries. Subsequently, considering the asymmetric input constraints, a discounted non-quadratic cost function is introduced. Moreover, in order to reduce controller updates, an event-triggered control law is developed for ADP algorithm. After that, to further simplify the complexity of controller design, this work is extended to a self-triggered case for relaxing the need for continuous signal monitoring by hardware devices. By employing the Lyapunov method, the uniform ultimate boundedness of all signals is proved to be guaranteed. Finally, a simulation example on a mass–spring–damper system subject to asymmetric input constraints is provided to validate the effectiveness of the proposed control scheme.