A numerical solution of open-loop Nash equilibrium in nonlinear differential games based on Chebyshev pseudospectral method

Abstract

In general, the applications of differential games for solving practical problems have been limited, because all calculations had to be done analytically. In this investigation, a simple and efficient numerical method for solving nonlinear nonzero-sum differential games with finite- and infinite-time horizon is presented. In both cases, derivation of open-loop Nash equilibria solutions usually leads to solving nonlinear boundary value problems for a system of ODEs. The proposed numerical method is based on a combination of minimum principle of Pontryagin and expanding the required approximate solutions as the elements of Chebyshev polynomials. Applying Chebyshev pseudospectral method, two-point boundary value problems in differential games are reduced to the solution of a system of algebraic equations. Finally, several examples are given to demonstrate the accuracy and efficiency of the proposed method and a comparison is made with the results obtained by fourth order Runge–Kutta method.