<p>The distributed optimal output synchronization problem for the leaderless heterogeneous multi-agent system with a general global cost function is investigated for the first time by linear quadratic (LQ) optimal control theory. Conventional algorithms for heterogeneous systems are quite complex, requiring the design of a virtual reference generator and the solving of regulation equations. This paper presents a novel distributed asymptotically optimal controller by incorporating the design of distributed observer and feedforward controller. A general form of the distributed controller is obtained by solving an augmented algebraic Riccati equation, which is parallel to classical optimal control theory. The optimal topology is an arbitrary directed graph containing only a spanning tree. It is shown that the proposed algorithms outperform the traditional consensus methods in the convergence speed by selecting proper observer gain matrices, and eliminate the reliance on the nonzero eigenvalues of Laplacian matrix. Simulation example further demonstrates the effectiveness of the proposed scheme and a faster superlinear convergence speed than the existing algorithm.</p>