This article proposes a novel optimal consensus protocol for a class of leaderless multiagent systems, where each agent is described by a second-order nonlinear system and agents interact with each other on switching networks. Any global information, including the eigenvalues of the Laplacian matrix, is unavailable in the control scheme development. The reference trajectory is designed for each agent, and the corresponding performance function, which reflects the off-track error evolution and control cost, is proposed. The sufficient condition for the synchronization of reference trajectories, which does not rely on the topology dwell time, is established by constructing an appropriate current topology-independent Lyapunov function. Due to the fact that the nonlinear function in the system dynamical equation of each agent is unknown, an equation termed integral reinforcement learning (IRL) equation is provided, and it is strictly proven that the provided IRL equation is equivalent to the given Hamilton–Jacobi–Bellman equation. The model-free optimal feedback control law is then derived based on the IRL technique. In the implementation of the developed control scheme, the neural network approximation tool is adopted, and the scheme is applied to a numerical system to show its effectiveness.