Multi-player nonzero-sum Nash differential game: variation and pseudo-spectral method

A novel method is presented to solve the nonzero‐sum multi‐player Nash differential game. It combines the use of the variation and Legendre pseudo‐spectral methods. By the variation method, the original game is converted into a regular optimal control problem, avoiding the need to solve the associated Hamilton–Jacobi equation. Then the latter problem is converted into a common nonlinear programming problem via the Legendre pseudo‐spectral method, by which the saddle‐point for the original game can be achieved accurately. As an illustration, the air combat between two pursuers and an evader is formulated as a nonzero‐sum differential game. The simulation results show that numerical solutions can converge to the saddle‐points from different initial conditions, which demonstrates the feasibility and validity of the proposed method. Because the solution process requires little computational time, this method will allow for the development of a real time air combat control strategy. In addition, the simulations show that if the initial states of the two pursuers are fixed, there is an optimal initial heading angle for the evader to delay the interception time most effectively.