The paper is concerned with the application of quadratic optimization for motion control to feedback control of robotic systems using neural networks. Explicit solutions to the Hamilton-Jacobi-Bellman (H-J-B) equation for optimal control of robotic systems are found by solving an algebraic Riccati equation. It is shown how neural networks can cope with nonlinearities through optimization with no preliminary off-line learning phase required. The adaptive learning algorithm is derived from Lyapunov stability analysis, so that both system tracking stability and error convergence can be guaranteed in the closed-loop system. The filtered tracking error or critic gain and the Lyapunov function for the nonlinear analysis are derived from the user input in terms of a specified quadratic performance index. Simulation results on a two-link robot manipulator show the satisfactory performance of the proposed control schemes even in the presence of large modeling uncertainties and external disturbances.