A numerical treatment based on Bernoulli Tau method for computing the open-loop Nash equilibrium in nonlinear differential games

The Tau method based on the Bernoulli polynomials is implemented efficiently to approximate the Nash equilibrium of open-loop kind in non-linear differential games over a finite time horizon. By this treatment, the system of two-point boundary value problems of differential game ex-tracted from Pontryagin’s maximum principle is transferred to a system of algebraic equations that Newton’s iteration method can be used for solving it. Also, for the mentioned approximation by the Bernoulli polynomials, the convergence analysis and the error upper bound are discussed. To demonstrate the applicably and accuracy of the proposed approach, some illustrated examples are presented at the final.