Robust optimal feedback control design for uncertain systems based on artificial neural network approximation of the Bellman's value function

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.