The theory of optimal control is applied to multi-stage (i.e., multiple-layered) neural-network (NN) learning for developing efficient second-order algorithms, expressed in NN notation. In particular, we compare differential dynamic programming, neighboring optimum control, and stagewise Newton methods. Understanding their strengths and weaknesses would prove useful in pursuit of an effective intermediate step between the steepest descent and the Newton directions, arising in supervised NN-learning as well as reinforcement learning with function approximators.