A computational method for finding feedback Nash equilibrium solutions (FBNES) in Nonzero-Sum Differential Games (NZSDG) Based on the Variational Iteration Method (VIM)

In this paper, we applied a semi-analytical approach called the Variational Iteration Method (VIM) to solve Nonzero-Sum Differential Games (NZSDGs). With this method, the Hamilton–Jacobi–Bellman (HJB) equation as a partial differential equation (PDE) is approximated iteratively using a correction function. This approach allows us to obtain approximations of the value functions, the feedback Nash equilibrium solutions (FBNES), and an estimation of the optimal performance index (PI). The proposed method provides a simple way to solve NZSDGs and is applied to obtain solutions for both linear-quadratic models in two scenarios and two applied nonlinear cases. Note that, to highlight their practical relevance, we have selected these models. The numerical results of the presented case studies demonstrate that the proposed method can generate approximate solutions with high accuracy. The convergence of the method, based on the Banach fixed point theorem (BFPT), is investigated and discussed in all examples. Additionally, a comparative analysis evaluates the efficiency and accuracy of the proposed method against the backward fourth-order Runge–Kutta (BRK4) method and the backward finite difference (BFD) scheme. Results show that VIM achieves similar or higher accuracy with significantly fewer iterations, while enhancing computational efficiency. Unlike the BRK4 and BFD methods, which rely on discretization, the VIM generates a continuous analytical solution, offering greater flexibility and improved accuracy for solving differential game problems.