This work extends and compares some recent model+learning-based methodologies for optimal control with input saturation. We focus on two methodologies: a model-based actor-critic (MBAC) strategy, and a nonlinear policy iteration strategy. To evaluate the performance of the algorithms, these strategies are applied to the swinging up an inverted pendulum. Numerical simulations show that the neural network approximation in the MBAC strategy can be poor, and the algorithm may converge far from the optimum. In the MBAC approach neither stabilization nor monotonic convergence can be guaranteed, and it is observed that the best value function is not always corresponding to the last one. On the other side the nonlinear policy iteration approach guarantees that every new control policy is stabilizing and generally leads to a monotonically decreasing cost.