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In this study, a local approximated solution for the Hamilton–Jacobi–Bellman equation based on differential neural networks is proposed. The approximated Value function is used to obtain a feedback control law for a class of uncertain dynamical systems. The approach to deal with the uncertainties of the dynamical system is a min–max method that yields obtaining a robust-like solution for an optimal control problem. The proposed cost functional is presented in the Bolza form and the necessary and sufficient conditions for getting the min–max optimal solution with the designed robust version of the dynamic programming approach are provided. Moreover, the effect of the neural network approximation is studied. All the analysis considers a smoothness assumption regarding the Value function. The practical stability of the system with the neural network feedback optimal control is formally demonstrated by means of the Lyapunov method. Finally, the performance of the control law is compared in simulation with a classical optimal controller. The proposed controller overcomes the results produced by the classical controller, which is confirmed by the functional evaluation.

State estimation for uncertain systems affected by external noises has been recognized as an important problem in control theory for either discrete and continuous plants. This paper deals with the state observation problem when the discrete-time dynamic model of a plant is partially unknown and it is affected by some sort of uncertainties and external perturbations. Recurrent Neural Networks (RNN) have shown several advantages to treat many different control and state estimation problems. In this paper, a new discrete-time Luenberger-like observer using the structure of a RNN is proposed. The class of discrete-time nonlinear system just has the input-output pairs as available information. The neural observer is training off-line using a class of least mean square method for matrix parameters. Lyapunov theory is employed to obtain the upper bounds for the weights dynamics as well for the estimation error and the learning laws to ensure the convergence of the observer. Simulation results using the van der Pol oscillator as data generator are presented to demonstrate the effectiveness of the proposed neural observer.