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This research is dedicated to developing a min–max robust control strategy for a dynamic game involving pursuers, evaders, and defenders in a multiple-missile scenario. The approach employs neural dynamic programming, utilizing multiple continuous differential neural networks (DNNs). The competitive controller devised addresses the robust optimization of a joint cost function that relies on the trajectories of the pursuer–evader–defender system, accommodating an uncertain mathematical model while adhering to control restrictions. The dynamic programming min–max formulation facilitates robust control by accounting for bounded modeling uncertainties and external disturbances for each game component. The value function of the Hamilton–Jacobi–Bellman (HJB) equation is approximated by a DNN, enabling the estimation of the closed-loop formulation for the joint dynamic game with state restrictions. The controller’s design is grounded in estimating the state trajectory under the worst possible uncertainties and perturbations, providing a robustness factor through the robust neural controller. The learning law class for the time-varying weights in the DNN is generated by studying the HJB partial differential equation for the missile motion for each player in the dynamic game. The controller incorporates the solution of the obtained learning laws and a time-varying Riccati equation, offering an online solution to the control implementation. A recurrent algorithm, based on the Kiefer–Wolfowitz method, adjusts the initial conditions for the weights to satisfy the final condition of the given cost function for the dynamic game. A numerical example is presented to validate the proposed robust control methodology, confirming the optimization solution based on the DNN approximation for Bellman’s value function. [ABSTRACT FROM AUTHOR]

This study focuses on developing a continuous differential neural network (DNN) approximating min–max robust control by applying dynamic neural programming. The suggested controller is applied to a class of nonlinear perturbed systems providing satisfactory dynamics for a given cost function depending on both the trajectories of the perturbed system and the designed constrained control actions. The min–max formulation for dynamic programming offers reliable control for restricted modeling uncertainties and perturbations. The suggested design considers control norm restrictions in the optimization problem. DNN's approximation of the worst (with respect to the admissible class of perturbations and uncertainties) value function of the Hamilton–Jacobi–Bellman (HJB) equation enables the estimation of the closed-loop formulation of the controller. The robust version of the HJB partial differential equation is studied to create the learning law class for the time-varying weights in the DNN. The controller employs a time-varying Lyapunov-like differential equation and the solution of the corresponding learning laws. A recurrent algorithm based on the Kiefer–Wolfowitz technique can be used by modifying the weights' initial conditions to fulfill the specified cost function's end requirements. A numerical example tests the robust control proposed in this study, validating the robust optimal solution based on the DNN approximation for Bellman's value function. • A robust optimal control is developed using approximate neural dynamic programming. • Robust optimal control is solved for nonlinear systems with restricted control actions. • A differential neural network is used to approximate Belman's function for Hamilton–Jacobi–Bellman equation. • The learning laws are derived based on the Hamilton–Jacobi–Bellman equation. [ABSTRACT FROM AUTHOR]