To address communication constraints for networked nonlinear multi-agent cooperative tracking, this paper develops a prescribed-performance-based self-triggered optimal tracking control algorithm, in which state constraints under arbitrarily bounded initial conditions are considered. First, a dynamic self-triggered prescribed-performance state observer is proposed, where the triggering threshold is adaptively regulated by the prescribed performance function and the tracking evolution. The observer guarantees convergence of the observation error within a prescribed time while significantly reducing communication load. Next, leveraging the estimated reference trajectory states, a prescribed-performance optimal tracking controller applicable to arbitrary bounded initial conditions is constructed. By employing nonlinear mappings, time-varying performance functions, and auxiliary compensation terms, the original constrained nonlinear multi-agent system is equivalently transformed into an unconstrained form. Based on the actor–critic reinforcement learning architecture, the optimal tracking controller is presented. Rigorous Lyapunov-based analysis establishes the stability of the closed-loop system and the satisfaction of prescribed performance constraints. Finally, the numerical simulations and hardware experiments validate the effectiveness and practicality of the proposed optimal tracking control scheme.